# Dynamic Erdős-Pósa listing

This page is an attempt to construct an exhaustive listing of results about the Erdős–Pósa property for graphs. Some data has been collected when preparing this article.

Note that **the papers listed on this page have not all been peer-reviewed** (yet). All of them are availlable online or have been published in a journal or conference proceedings with peer-review.

If you notice a missing result, a typo, a broken link, or an outdated reference, please contact me and I will update the list.

MathJax is used for rendering formulas, so you need to have JavaScript enabled in order to see them.

### Quick links

- positive results: acyclic patterns, triangles,
**cycles**, cycles with length constraints, cycles with prescribed vertices,**minors**, topological minors, immersions, induced patterns, patterns with prescribed positions, classes with bounded parameters, fractional packings; - negative results: paths, cycles, minors, topological minors, immersions, induced patterns.

### Basic definitions

Let $\mathcal{H}$ be a class of graphs and let $G$ be a graph.

- An
*$\mathcal{H}$-vertex-packing*in $G$ is a collection of vertex-disjoint subgraphs of $G$, each isomorphic to a member of $\mathcal{H}$. - An
*$\mathcal{H}$-vertex-cover*in $G$ is a set $X$ of vertices of $G$ such that $G\setminus X$ has no subgraph isomorphic to a member of $\mathcal{H}$.

Let $\mathcal{G}$ be a class of graphs. We say that $\mathcal{H}$ (refered to as the *guest class*) has the *vertex-Erdős–Pósa property* in the class $\mathcal{G}$ (the *host class*) with *gap* $f$ if, for every $G \in \mathcal{G}$ and $k \in \mathbb{N}$,

- either $G$ has an $\mathcal{H}$-vertex-packing of size $k+1$;
- or $G$ has a $\mathcal{H}$-vertex-covering of size $f(k)$.

The notions of *$\mathcal{H}$-edge-packing*, *$\mathcal{H}$-edge-cover*, and *edge-Erdős–Pósa property* can be defined similarly by replacing “vertex” by “edge” in the definitions above.

**Notice that a different definition of the gap (with $k+1$ replaced by $k$ above) is sometimes used in the litterature (but not on this page), leading to slightly different values of the gap.**

The fourth column (T.) of the tables below refers to the type of Erdős–Pósa property: *v* for vertex and *e* for edge. The definitions of the other variants of the Erdős–Pósa property mentioned below (*w* for weighted, *v _{1/2}* for vertex half-integral, etc.) are not given here but can be found in the corresponding papers.

Notation of the type $O_t(k)$ and $o_t(k)$ is used to stress that the hidden constants depend on $t$.

## Positive results

### Acyclic patterns

Ref. | Guest class | Host class | T. | Gap at most | Remarks |
---|---|---|---|---|---|

[Kőn31] | $K_2$ | bipartite | v | $k$ | |

[LY78] | directed cuts | any digraph | e | $k$ | |

[Lov76] | directed cuts | any digraph | e | $k$ | |

[Men27] | $(S,T)$-paths | any | v/e | $k$ | |

[MNL84] | $(S,T)$-paths of length $\geq t$ | any | v | $(3(t + 2) - 5)k$ | |

[HU19] | $(S,T)$-paths of length $\geq t$ | any | e | unspecified | |

[Grü38] | directed $(S,T)$-paths | any digraph | v/e | $k$ | |

[Gal64] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ |
any | v | $2k$ | |

[GV95] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$, $S$ independent |
graphs where every block has $\leq 2$ cutvertices and is either a cycle or a clique | v | $k$ | |

[Sam92] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$, $S$ independent |
unicyclic graphs whose cycle has $\leq 2$ vertices of degree $\geq 3$ | v | $k$ | |

[Sam83] [TV89] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$, $S$ independent |
trees | v | $k$ | |

[KKKX20] | directed odd $S$-paths$S\subseteq V(G)$ and a directed odd $S$-path is a directed path of odd length with both endpoints in $S$ |
any | v | $k$ | |

[Mad78b] | $\mathcal{S}$-paths$\mathcal{S}$ is a collection of disjoint subsets of $V(G)$ and a $\mathcal{S}$-path is a path connecting different sets in $\mathcal{S}$ |
any | v | unspecified | see [Sch01] |

[Mad78a] | $\mathcal{S}$-paths$\mathcal{S}$ is a collection of disjoint subsets of $V(G)$ and a $\mathcal{S}$-path is a path connecting different sets in $\mathcal{S}$ |
any | e | $2k$ | see [SS04], [BHJ18a] |

[HU19] | $\mathcal{S}$-paths$\mathcal{S}$ is a collection of disjoint subsets of $V(G)$ and a $\mathcal{S}$-path is a path connecting different sets in $\mathcal{S}$ of length $\geq t$ |
any | e | unspecified | |

[GGR+09] | odd $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ |
any | v | $2k$ | |

[BHJ18a] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length at least $t$ |
any | v | $4kt$ | |

[HU19] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $\geq t$ |
any | e | unspecified | |

[BHJ18a] | even $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ |
any | v | $10k$ | |

[BU18] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $0 \mod 4$ |
any | v | unspecified | |

[BU18] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $2 \mod 4$ |
any | v | unspecified | |

[Ulm20] | $S$-paths of length $0 \mod m$, where $m$ is an odd prime | any | v | unspecified | |

[Ulm20] | $S$-paths of length $d \mod m$ if $\forall x,y \in \{0, \dots, m-1\}, y \neq 0 \Rightarrow \exists n \in \mathbb{Z}, 2x+ny = d \mod m$ | any | v | unspecified | |

[CGG+06] | non-zeroAn undirected path in a digraph is non-zero if the sum of the labels of its arcs (with signs depending on the orientation of the edge) is non-zero. $S$-paths |
edge-group-labeled digraphs | v | $2k$ | |

[Wol10] | non-zeroA path is non-zero if the sum of the labels of its edges is non-zero. $S$-paths |
edge-group-labeled graphs | v | $50 k^4$ | |

[Ulm20] | zero $S$-paths | graphs whose edges are labeled by a finite abelian group | v | unspecified | |

[Ulm20] | $S$-paths of length $\geq t$ and value $\gamma$, where $\gamma \in \Gamma$ | graphs whose edges are labeled by an abelian group $\Gamma$ such that $\forall x,y \in \Gamma, y \neq 0 \Rightarrow \exists n \in \mathbb{Z}, 2x + ny = \gamma$ | v | unspecified | |

[IS17] | odd $(u,v)$-trailsAn $(u,v)$-trail is a walk from $u$ to $v$ that may have repeated vertices but no repeated edge. A trail is odd if it has an odd number of edges. |
any | e | $2k+1$ | |

[MW15] | $H$-valid paths, $H$ with no matching of size $t$ | any | v | $2^{2^{O(k+t)}}$ | |

[FJW13] | $\mathcal{M}(H)$, $H$ forest | any | v | $O_{H}(k)$ | |

[BHJ18a] | $S$-$t$-combsAn elementary $t$-comb can be obtained by adding a pendant vertex to each internal vertex of a path on $t$ edges. A $t$-comb is any subdivision of an elementary $t$-comb. Let $S\subseteq V(G)$. A $S$-$t$-comb is a $t$-comb whose leaves, and only the leaves, belong to $S$. |
any | v | $4kt$ | |

[SU20] | $S$-$t$-trees$A$ is a vertex set, $t\in \mathbb{N}$ and a $S$-$t$-tree is a tree that contain $m$ vertices of $A$. |
any | e | $2t^2k^2$ |

### Triangles

The following table contains results related to Tuza’s conjecture, that can be seen as Erdős-Pósa type problems.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[Tuz90] | triangles | planar graphs | e | $2k$ |

[Tuz90] | triangles | graphs with $m \geq 7n^2/16$ | e | $2k$ |

[Tuz90] | triangles | tripartite graphs | e | $7k/3$ |

[Kri95] | triangles | $\mathcal{T}(K_{3,3})$$\mathcal{T}(H)$ is the class of graphs containing a subdivision of $H$-free graphs | e | $2k$ |

[HK88] | triangles | tripartite graphs | e | $1.956k$ |

[Hax99] | triangles | any | e | $(3-\frac{3}{23})k$ |

[ALBT11] | triangles | odd-wheel-free graphs | e | $2k$ |

[ALBT11] | triangles | 4-colorable graphs | e | $2k$ |

[HKT11] | triangles | $K_4$-free planar graphs | e | $3k/2$ |

[HKT11] | triangles | $K_4$-free flatA graph is flat if every edge belongs to at most two triangles. graphs |
e | $3k/2$ |

[Pul15] | triangles | graphs with $\mathrm{mad} < 7$ | e | $2k$ |

[LBT16] | triangles | graphs whose triangle graphThe triangle graph of $G$ is the graph whose vertices are triangles of $G$ and where two edges are adjacent if the corresponding triangles share an edge. is perfect |
e | $2k$ |

[Tuz94] | directed triangles | planar oriented graphs | e | $k$ |

[MPT18] | directed triangles | directed graphs | e | $2k-1$ |

### Cycles

For cycles with length or modularity constraints or using prescribed vertices ($S$-cycles), see the next tables.

Ref. | Guest class | Host class | T. | Gap at most | Remarks |
---|---|---|---|---|---|

[Lov65] | cycles | graphs without 2 vertex-disjoint cycles | v | =3 | |

[Vos69] | cycles | graphs without 3 vertex-disjoint cycles | v | =6 | |

[Vos67] | cycles | graphs without 4 vertex-disjoint cycles | v | 12 | see [Vos69] |

[EP65] | cycles | any | v | $O(k\log k)$ | |

[Sim67] | cycles | any | v | $\left (4 + o(1)\right )k \log k$ | |

[Vos69] | cycles | any | v | $\left (2 + o(1) \right )k \log k$ | |

[Vos69] | cycles | any | v | $4k \left (1+ \log \left (k + \frac{3}{2} \right ) \right)$ | |

see [Die05] | cycles | any | e | $(2 + o(1))k \log k$ | |

[DZ02] | cycles | weighted graphs with no induced subdivision of $K_{2,3}$, a wheel, or an odd ringAn odd ring is a graph obtained from an odd cycle by replacing every edge ${u,v}$ by either a triangle containing ${u,v}$, or by two triangles on vertices ${u,a,b}$ and ${v,c,d}$ together with the edges ${b,c}$ and ${a,d}.$ |
w | $k$ | |

[DXZ03] | cycles | graphs with no induced subdivision of $K_{3,3}$, a wheel, or an odd ringAn odd ring is a graph obtained from an odd cycle by replacing every edge ${u,v}$ by either a triangle containing ${u,v}$, or by two triangles on vertices ${u,a,b}$ and ${v,c,d}$ together with the edges ${b,c}$ and ${a,d}.$ |
v | $k$ | |

[BD92] | cycles | planar graphs | v | $54k$ | |

[KLL02] | cycles | planar graphs | v | $5k$ | |

[CHC12] [MYZ13][CGH14] | cycles | planar graphs | v | $3k$ | |

[CGH14] | cycles | graphs that embed in a closed surface of Euler characteristic $\geq 0$ | v | $3k$ | |

[CGH14] | cycles | graphs that embed in a closed surface of Euler characteristic $c \leq 0$ | v | $3k - 103c$ | |

[KLL02] | cycles | outerplanar graphs | v | $2k$ | |

[Mun16] | cycles | $(K_4, \text{claw}, \text{diamond})$-free graphs | v | $2k$ | |

[Mun16] | cycles | planar claw-free graphs with $\Delta\leq 4$ | v | $2k$ | |

[BDM+19] | cycles | planar subcubic graphs | v | $2k$ | |

[MYZ13] | cycles | planar graphs | e | $4k-1$ | |

[BD92] | faces | plane graphs | v | $27k$ | |

[BD92] + [MYZ13] | faces | plane graphs | v | $6k$ | |

[BD92] + [MYZ13] + [GR09] | cycles or facesWhen one is looking for connected covers in planar graphs, covering faces is equivalent to covering cycles (see Lemma 1 in [GR09]). | connected planar graphs with $\delta \geq 3$ | v | $66k - 14$ | the cover additionally induces a connected subgraph |

[RRST96] | directed cycles | any digraph | v | unspecified | |

[MMP+19] | directed cycles | any digraph | v_{1/4} |
$O(k^4)$ | |

[MMP+19] | directed cycles | any digraph | v_{1/2} |
$O(k^6)$ | |

[RS96] | directed cycles | planar digraphs | v | $O(k\log(k)\log\log k)$ | |

[RS96] + [GW96] | directed cycles | planar digraphs | v | $63k$ | |

[STV22] | directed cycles | planar digraphs | v | $12k$ | |

[GT11] | directed cycles | strongly planar digraphsA digraph is strongly planar if it has a planar drawing such that for every vertex $v$, the edges with head $v$ form an interval in the cyclic ordering of edges incident with $v$ (definition from [GT11]). |
v | $k$ | |

[GT11] | directed cycles | digraphs with no butterfly minorSee [GT11] for a definition. isomorphic to an odd double circuitAn odd double circuit is a digraph obtained from an undirected circuit of odd length more than 2 by replacing each edge by a pair of directed edges, one in each direction. or $F_7$$F_7$ is the digraph obtained from the directed cycle on vertices $v_1, \dots, v_7,v_1$, by adding the edges creating the directed cycle $v_1,v_3, v_5, v_7, v_2, v_4, v_6, v_1$. |
v | $k$ | |

[LY78] | directed cycles | planar digraphs | e | $k$ | |

[Sey96] | directed cycles | eulerian digraphs with a linkless embedding in 3-space | e | $k$ | |

[HJW18] | cycles non homologous to zero | embedded graphs | v_{1/2} |
unspecified | |

[STV22] | any collection of uncrossable cycles | planar graphs or planar digraphs | v | $12k$ | |

[STV22] | any collection of uncrossable cycles | planar graphs or planar digraphs | e | $9.6k$ |

### Cycles with length constraints

Includes cycles of a certain value with respect to a labeling of the edges or vertices of the graph.

Ref. | Guest class | Host class | T. | Gap at most | Remarks |
---|---|---|---|---|---|

[Ree99] | odd cycles | planar graphs | v | superexponential | |

[FHRV06] | odd cycles | planar graphs | v | $10k$ | |

[KSS12] | odd cycles | planar graphs | v | $\max(0, 6k-2)$ | |

[Tho01] | odd cycles | $2^{3^{9k}}$-connected graphs | v | $2k$ | |

[RR01] | odd cycles | $576k$-connected graphs | v | $2k$ | |

[KR09] | odd cycles | $24k$-connected graphs | v | $2k$ | |

[Ree99] | odd cycles | $k$-near bipartiteA graph is $k$-near bipartite if every set of $p$ vertices contains a stable set of size at least $p/2 - k$. graphs |
v | unspecified | |

[KN07] | odd cycles | embeddable in an orientable surface of Euler genus $t$ | v/e | unspecified | |

[BR00] | odd cycles | planar | e | unspecified | |

[KV04] [FHRV06] | odd cycles | planar graphs | e | $2k$ | |

[KK16] | odd cycles | 4-edge-connected graphs | e | $2^{2^{O(k \log k)}}$ | |

[Ree99] | odd cycles | any | v_{1/2} |
unspecified | |

[BHJ18a] | even cycles | any | e | $O(k^2 \log k)$The function obtained in [BHJ18a] is $k \mapsto 6kh(k)$, where $h$ is the function for the vertex Erdős-Pósa property of even cycles. It was later proved that $h(k) = O(k \log k)$ (see below, stated for cycles of length $0 \mod t$). | |

[CJU19] | even cycles | any | e | $O(k \log k)$ | |

[Tho88] | cycles of length $0 \mod t$ | any | v | $2^{t^{O(k)}}$ | |

[CC13] | cycles of length $0 \mod t$ | any | v | $k~ \mathrm{polylog}~k \cdot 2^{\mathrm{poly}(t)}$ | |

[CHJR18] | cycles of length $0 \mod t$ | any | v | $O_t(k \log k)$ | |

[CHJR18] | cycles of length $0 \mod t$ | any proper minor-closed class $\mathcal{F}$ | v | $O_{t,\mathcal{F}}(k)$ | |

[KW06] | non-zero cycles | $(15k/2)$-connected group-labeled graphs | v | $2k$ | |

[Wol11] | non-zero cycles | group-labeled graphs, c.f. [Wol11] | v | $c^{k^{c’}}$ for some $c,c’$ | |

[Wol11] | cycles of non-zero length mod $2t+1$ | any | v | $c^{k^{c’}}$ for some $c,c’$ | |

[HJW18] | doubly non-zero cycles, c.f. [HJW18] | doubly group-labeled graphs | v_{1/2} |
unspecified | |

[HJW18] | odd cycles non homologous to zero | embedded graphs | v_{1/2} |
unspecified | |

[GHK+21] | cycles belonging to certain $\mathbb{Z}_2$-homotopy classes of $\Sigma$ | graphs embedded on a surface $\Sigma$ | v_{1/2} |
unspecified | |

[GHK+21] | cycles whose valueThe value of a cycle (with respect to an edge-labeling with the elements of some group) is the sum of the labels of its edges. equals a certain value | graphs whose edges are labeled by a finite abelian group | v_{1/2} |
unspecified | |

[GHK+21] | cycles whose valuesThe value of a cycle (with respect to an edge-labeling with the elements of some group) is the sum of the labels of its edges. avoid a finite number of valuesFor each of the groups used in the labeling, a finite number of values to avoid is fixed at the beginning. | graphs whose edges are labeled with finitely many abelian groups | v_{1/2} |
unspecified | besides $k$, the gap only depends on the number of groups and of values to avoid |

[GHK+21] | cycles whose valuesThe value of a cycle (with respect to a vertex-labeling with the elements of some group) is the sum of the labels of its vertices. avoid a finite number of valuesFor each of the groups used in the labeling, a finite number of values to avoid is fixed at the beginning. | graphs whose vertices are labeled with finitely many abelian groups | v_{1/2} |
unspecified | besides $k$, the gap only depends on the number of groups and of values to avoid |

[Bir03] | cycles of length $\geq 4$ | graphs without 2 vertex-disjoint cycles of length $\geq 4$ | v | =4 | |

[Bir03] | cycles of length $\geq 5$ | graphs without 2 vertex-disjoint cycles of length $\geq 5$ | v | =5 | |

[BBR07] | cycles of length $\geq t$ | any | v | $(13+o_t(1))tk^2$ | |

[FH14] | cycles of length $\geq t$ | any | v | $(6t+4+o_t(1))k\log k$ | |

[MNŠW17] | cycles of length $\geq t$ | any | v | $6kt + (10 + o(1)) k \log k$ | |

[BHJ16] | cycles of length $\geq t$ | any | e | $O(k^2 \log k + kt)$ | |

[CJU19] | cycles of length $\geq t$ | any | e | $8k(t−1)(\log_2(kt) + 1)$ | |

[KKKX20] | directed odd cycles | any digraph | v_{1/2} |
unspecified | |

[HM13] | directed cycles of length $\geq 3$ | any digraph | v | unspecified | |

[AKKW16] | directed cycles of length $\geq t$ | any digraph | v | unspecified |

### Cycles with prescribed vertices or edges

For a set $S \subseteq V(G)$, an *$S$-cycle* of $G$ is a cycle of $G$ containing at least one vertex of $S$. The results listed in this section relate, for any $S \subseteq V(G)$, the maximum number of $S$-cycles with the minimum number of vertices whose removal destroys all $S$-cycles. This setting extends to cycles that of $S$-paths mentioned in the acyclic patterns section.

Ref. | Guest class | Host class | T. | Gap at most | Remarks |
---|---|---|---|---|---|

[KKM11] | $S$-cycles | any | v | $40k^2 \log k$ | |

[PW12] | $S$-cycles | any | v/e | $O(k \log k)$ | |

[KKL19] | $S$-cycles | graphs without two vertex-disjoint $S$-cycles | v | =4 | |

[BJS17] | $S$-cycles of length at least $t$ | any | v | $O(tk\log k)$ | |

[Joo14] | odd $S$-cycles | $50k$-connected graphs | v | $O(k)$ | |

[KK13] | odd $S$-cycles | any | v_{1/2} |
unspecified | |

[Bru19] | even $S$-cycles | any | e | unspecified | |

[KK12] | directed $S$-cycles | all digraphs | v_{1/5} |
unspecified | |

[KKKK13] | odd directed $S$-cycles | any digraph | v_{1/2} |
unspecified | |

[HJW18] | $(S,S’)$-cycles$S,S’ \subseteq V(G)$ and an $(S,S’)$-cycle is a cycle that contains vertices of both $S$ and $S’$ |
any | v | unspecified | |

[GHK+21] | cycles that intersect at least $t_i\leq t$ times the set $S_i$ for every $i \in \{1, \dots, m\}$, for a collection $S_1, \dots, S_m$ of subsets of vertices of the graph | any | v_{1/2} |
unspecified | besides $k$, the gap only depends on $m$ and $t$ |

[STV22] | $(=1)$-edge-$S$-cyclesHere $S$ is a set of edges and we consider those cycles that contain exactly one edge from $S$. | planar graphs | v | 12 |

### Minors

For every graph $H$, we denote by $\mathcal{M}(H)$ the class of graphs that can be contracted to~$H$. (So $H$ is a minor of $G$ iff some subgraph of $G$ is isomorphic to a graph in $\mathcal{M}(H)$.) For every digraph $D$, we denote by $\vec{\mathcal{M}}_b(D)$ (respectively $\vec{\mathcal{M}}_s(D)$) the class of all digraphs that contain $D$ as a butterfly contraction (respectively strong contraction).

For every $t \in \mathbb{N}$, $\theta_t$ is the multigraph with two vertices connected by one edge of multiplicity $t$.
For every $t\in \mathbb{N}$, a digraph is said to be *$t$-semicomplete* if for every vertex $v$ there are at most $t$ vertices that are not connected to $v$ by an arc (in either direction). A *semicomplete* digraph is a 0-semicomplete digraph.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[Sim67] | $\mathcal{M}($complement of ($K_{3,3}$ minus an edge)) | any | v | $(4000+o(1))k \log k$ |

[RS86] | $\mathcal{M}(H)$, $H$ planar | any | v | unspecified |

[RS86] | $\mathcal{M}(H)$, $H$ planar with $q$ connected components | ${G, \mathbf{tw}(G)\leq t}$ | v | $(t-1)(kq-1)$ |

[FST11] | $\mathcal{M}(H)$, $H$ planar connected | any proper minor-closed class $\mathcal{F}$ | v | $O_{H,\mathcal{F}}(k)$ |

[DKW12] | $\mathcal{M}(K_t)$ | $O(kt)$-connected graphs | v | unspecified |

[FLM+13] | $\mathcal{M}(\theta_t)$ | any | v | $O(t^2k^2)$ |

[RT13] | $\mathcal{M}(H), \mathbf{pw}(H) \leq 2$ and $H$ connected on $h$ vertices | any | v | $2^{O(h^2)}\cdot k^2\log k$ |

[CC13] + [CC14] | $\mathcal{M}(H)$, $H$ planar connected on $h$ vertices | any | v | $\mathrm{poly}(h)\cdot k\cdot \mathrm{polylog}~k$ |

[RST16] | $\mathcal{M}(\theta_t)$ | any | e | $k^2t^2 \mathrm{polylog}~kt$ |

[RST16] | $\mathcal{M}(\theta_t)$ | any | e | $k^4t^2 \mathrm{polylog}~kt$ |

[SU20] | $\mathcal{M}(H)$ where $H$ is the $2 \times 3$ grid | any | e | $O(k^3 \log k)$ |

[SU20] | $\mathcal{M}(\text{house graph})$ | any | e | $O(k^2 \log k)$ |

[AKKW16] | $\vec{\mathcal{M}}_{b}(H)$, $H$ is a butterfly minor of a cylindrical directed gridIntuitively: concentric cycles directed in the same direction + directed paths between the innermost cycle and the outermost one, in alternating directions. See [AKKW16] for a formal definition. | any digraph | v | unspecified |

[CRST17] | $\mathcal{M}(H)$, $H$ connected | ${G, \mathbf{tpw}(G) \leq t}$ | v/e | $O_{H,t}(k)$ |

[CRST17] | $\mathcal{M}(\theta_t)$ | any | v/e | $O_t(k\log k)$ |

[CRST17] | $\mathcal{M}(\theta_{t,t’})$$\theta_{t,t’}$ is the multigraph obtained from \(\theta_t \cup \theta_{t'}\) by identifying one vertex of \(\theta_t\) with one vertex of \(\theta_{t'}\). | simple graphs | e | unspecified |

[BH17] | $\mathcal{M}(K_4)$ | any | e | $O(k^8 \log k)$ |

[AFH+17] | $\mathcal{M}(W_t)$$W_t$ is the wheel on $t+1$ vertices, i.e. the graph obtained by adding a dominating vertex to a cycle of length $t$., $t \in \mathbb{N}$ | any | v | $O_t(k \log k)$ |

[Ray18] | \(\vec{\mathcal{M}}_{s}(H)\)\(\vec{\mathcal{M}}_{s}(H)\) is the class of strong minors of \(H\), i.e. digraphs that can be obtained from \(H\) after contracting strongly-connected subgraphs. | tournaments | v | unspecified |

[Ray18] | $\vec{\mathcal{M}}_{b}(H)$, $H$ strongly-connected | tournaments | v | unspecified |

[CHJR18] | $\mathcal{M}(H)$, $H$ planar | any | v | $O_H(k \log k)$ |

[BJS18] | $\mathcal{M}(H)$, when $H$ belongs to a proper subclassSee the paper for the details and the definition of minors in labeled graphs. of labelled 2-connected planar graphs | any | v | unspecified |

[KM19] | graphs of $\mathcal{M}(H)$ meeting $\geq \ell$ of the prescribed subsets, for $\ell \in \mathbb{N}_{\geq 1}$ and $H$ a planar graph with $\geq \ell-1$ connected components (detailsThe host graph $G$ comes with a number of prescribed vertex subsets and the guest graphs are those subgraphs of $G$ that can be constracted to $H$ and intersect $\geq \ell$ of the prescribed subsets. The gap function does not depend on the number of prescribed subsets of the host graph.) | any graph with prescribed subsets | v | unspecified |

### Topological minors

For every graph $H$, we denote by $\mathcal{T}(H)$ the class of graphs that contain $H$ as a topological minor. For every digraph $D$, we denote by $\vec{\mathcal{T}}(D)$ the class of all digraphs that contain $D$ as a directed topological minors.

Let $H$ be a graph of maximum degree 3. As a graph $G$ contains $H$ as a minor iff it contains it as a topological minor, all the results stated in the previous section hold for topological minors if the guest graph has maximum degreee 3 (for instance $\mathcal{T}(H)$ has the Erdős-Pósa property for every planar $H$ with maximum degree 3, by [RS86]). These results are not restated here.

Ref. | Guest class | Host class | T. | Gap at most | Remark |
---|---|---|---|---|---|

[Tho88] | \(\mathcal{T}_{(0 \mod t)}(H)\)\(\mathcal{T}_{(0 \mod t)}(H)\) is the class of subdivisions of $H$ where every edge is subdivided into a path on $0 \mod t$ edges., $H$ connected planar subcubic | any | v | unspecified | |

from [CC13] | \(\mathcal{T}_{(0 \mod t)}(H)\)\(\mathcal{T}_{(0 \mod t)}(H)\) is the class of subdivisions of $H$ where every edge is subdivided into a path on $0 \mod t$ edges., $H$ connected planar subcubic | any | v | $O_{H,t}(k~ \mathrm{polylog}~k)$ | |

[CHJR18] | \(\mathcal{T}_{(0 \mod t)}(H)\)\(\mathcal{T}_{(0 \mod t)}(H)\) is the class of subdivisions of $H$ where every edge is subdivided into a path on $0 \mod t$ edges., $H$ planar subcubic | any | v | $O_{H,t}(k \log k)$ | |

[CHJR18] | \(\mathcal{T}_{(0 \mod t)}(H)\)\(\mathcal{T}_{(0 \mod t)}(H)\) is the class of subdivisions of $H$ where every edge is subdivided into a path on $0 \mod t$ edges., $H$ planar subcubic | any proper minor-closed class $\mathcal{F}$ | v | $O_{H, \mathcal{F},t}(k)$ | |

[AKKW16] | $\vec{\mathcal{T}}(H)$, $H$ is a topological minor of a cylindrical directed gridIntuitively: concentric cycles directed in the same direction + directed paths between the innermost cycle and the outermost one, in alternating directions. See [AKKW16] for a formal definition. | any digraph | v | unspecified | |

[Liu17] | $\mathcal{T}(H)$ | any | v_{1/2} |
unspecified | holds for rooted subdivisions |

[Ray18] | $\vec{\mathcal{T}}(H)$, $H$ strongly-connected | tournaments | v | unspecified | |

[BP21] | $\vec{\mathcal{T}}(H)$ | tournaments | v | $O_H(k \log k)$ |

### Immersions

For every graph $H$, we denote by $\mathcal{I}(H)$ the class of graphs that contain $H$ as an immersion.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[Liu15] | $\mathcal{I}(H)$ | 4-edge-connected | e | unspecified |

[Liu15] | \(\mathcal{I}_{1/2}(H)\)A graph $H$ is a half-integral immersion of a graph $G$ is $H$ is an immersion of the graph obtained by $G$ after duplicating the multiplicity of all its edges and \(\mathcal{I}_{1/2}(H)\) denotes the class of all graphs containing $H$ as a half-integral immersion. |
any | e_{1/2} |
unspecified |

[GKRT17] | $\mathcal{I}(H)$, $H$ planar subcubic connected on $h>0$ edges | any | e | $\mathrm{poly}(h \cdot k)$ |

[GKRT17] | $\mathcal{I}(H)$, $H$ connected on $h>0$ edges | ${G, \mathbf{tcw}(G) \leq t}$ | e | $h \cdot t^2 \cdot k$ |

[GKRT17] | $\mathcal{I}(H)$, $H$ connected on $h>0$ edges | ${G, \mathbf{tpw}(G) \leq t}$ | e | $h \cdot t^2 \cdot k$ |

[KK18a] | $\mathcal{I}(H)$ | 4-edge-connected | e | unspecified |

[Ray18] | $\vec{\mathcal{I}}(H)$, $H$ strongly-connected | tournaments | e | unspecified |

[BP21] | $\vec{\mathcal{I}}(H)$ | tournaments | e | $O_H(k^3)$ |

### Induced patterns

In this setting, a *$\mathcal{H}$-vertex-packing* (*$\mathcal{H}$-edge-packing*) in $G$ is a collection of vertex-disjoint (edge-disjoint) **induced** subgraphs of $G$.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[EP65] | cycles of length $\geq 3$ | any | v | $O(k\log k)$ |

[KK18b] | cycles of length $\geq 4$ | any | v | $O(k^2\log k)$ |

trivial | $\mathcal{T}(H)$, $H$ linear forest | any | v | $O(k)$ |

[ALM+18] | {cycles of length $\geq 4$} $\cup$ {asteroidal triples} | any | v | $O(k^2 \log k)$ |

[KR18] | $\mathcal{T}(H)$, $H\in \{\text{diamond}, \text{1-pan}, \text{2-pan}\}$ | any | v | $\mathrm{poly}(k)$ |

[Wei18] | cycles of length $\geq t$ for $t \geq 3$ | $K_{s,s}$-subgraph free graphs | v | unspecified |

### Patterns with prescribed positions

The setting of the results in this section is slightly different: instead of packing/covering all possible occurences of a given pattern in a host graph, we focus on a given list of possible occurences. For instance, one can consider a family $\mathcal{F}$ of (non necessarily disjoint) subtrees of a tree $T$, and compare the maximum number of disjoint elements in $\mathcal{F}$ with the minimum number of vertices/edges of $T$ intersecting all elements of $\mathcal{F}$. This is stressed in the following table by refering to guest graphs with words related to substructures (like “subtrees”). Note that there may be two isomorphic subtrees $F,F’$ of $T$ such that $F \in \mathcal{F}$ and $F’ \not \in \mathcal{F}$.

For every positive integer $t$, a *$t$-path* is a disjoint union of $t$
paths, and a *$t$-subpath* of a $t$-path $G$ is a subgraph that has a
connected intersection with every connected component of $G.$ The
concepts of *$t$-forests* and *$t$-subforests* are defined similarly.
We denote by $\textbf{cc}(G)$ the number of connected components of the graph $G$.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[HS58] | subpaths | paths | v | $k$ |

[GL69] | $t$-subpaths | $t$-paths | v | $O(k^{t!})$ |

[GL69] | subgraphs $H$ with $\textbf{cc}(H) \leq t$ | paths | v | unspecified |

[GL69] | $t$-subforests | $t$-forests | v | unspecified |

[GL69] | subtrees | trees | v | $k$ |

[Kai97] | $t$-subpaths | $t$-paths | v | $(t^2-t+1)k$ |

[Alo98] | $t$-subpaths | $t$-paths | v | $2t^2k$ |

[Alo02] | subgraphs $F$ with $\textbf{cc}(F) \leq t$ | trees | v | $2t^2k$ |

[Alo02] | subgraphs $H$ with $\textbf{cc}(H) \leq t$ | ${G, \textbf{tw}(G)\leq w}$ | v | $2(w+1)t^2k$ |

[Tuz94] | directed subtriangles | planar oriented graphs | e | $k$ |

### Classes with bounded parameters

Some other results on classes with bounded parameters appear in other tables.

Ref. | Guest class | Host class | T. | Gap at most |
---|---|---|---|---|

[RS86] | $\mathcal{M}(H)$, $H$ planar with $q$ connected components | ${G, \mathbf{tw}(G)\leq t}$ | v | $(t-1)(kq-1)$ |

[Tho88] | any family of connected graphs | ${G, \mathbf{tw}(G)\leq t}$ | v | $k(t+1)$ |

[FJW13] | ${H, \textbf{pw}(H) \geq t}$ | any | v | $O_t(k)$ |

[GKRT17] | $\mathcal{I}(H)$, $H$ connected on $h>0$ edges | ${G, \mathbf{tcw}(G) \leq t}$ | e | $h \cdot t^2 \cdot k$ |

[GKRT17] | $\mathcal{I}(H)$, $H$ connected on $h>0$ edges | ${G, \mathbf{tpw}(G) \leq t}$ | e | $h \cdot t^2 \cdot k$ |

[CRST17] | any finite family of connected graphs | ${G, \textbf{tpw}(G) \leq t}$ | v/e | $O_{t}(k)$ |

[MMP+19] | all digraphs with directed treewidth $\geq t$ | any digraph | v_{1/4} |
$\textrm{poly}(k,t)$ |

### Fractional packings

The results in this table relate the value of the (usual) packing / covering numbers with fractional packing / covering numbers (see the referenced papers for a definition).

Ref. | Guest class | Host class | $\nu^*$ | $\tau$ | Relationship |
---|---|---|---|---|---|

[Sey95] | directed cycles | any digraph | max fractional vertex packing | min integral vertex covering | $\tau \leq 4 \nu^* \ln(4 \nu^* ) \ln\log(4\nu^*)$ |

## Negative results

### Lower bounds for paths

Ref. | Guest class | Host class | T. | Gap at least |
---|---|---|---|---|

[MW15] | $H$-valid paths, $H$ with no matching of size $t$ | all graphs | v | unavoidable dependency in $t$ |

[BHJ18a] | $S$-paths of length $p \mod t$, for $t>4$ composite and fixed $p \in \{0, \dots, t-1\}$ | all graphs | v/e | arbitrary |

[BU18] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $1 \mod 4$ |
all graphs | v | arbitrary |

[BU18] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $3 \mod 4$ |
all graphs | v | arbitrary |

[BHJ18a] | odd/even $S$-$T$-paths$S,T\subseteq V(G)$ and a $S$-$T$-path is a path with an endpoint in $S$ and the other one in $T$. |
all graphs | v/e | arbitrary |

[BHJ18a] | odd/even $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ |
all graphs | e | arbitrary |

[BHJ18a] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $0 \mod 4$ |
all graphs | e | arbitrary |

[BHJ18a] | $S$-paths$S\subseteq V(G)$ and a $S$-path is a path with both endpoints in $S$ of length $0 \mod p$, for any prime $p$ |
all graphs | e | arbitrary |

[IS17] | odd $(u,v)$-trailsAn $(u,v)$-trail is a walk from $u$ to $v$ that may have repeated vertices but no repeated edge. A trail is odd if it has an odd number of edges. |
all graphs | e | $\geq 2k+1$ |

[BHJ18a] | $S$-$T$-$S$-paths$S.T \subseteq V(G)$. A $S$-$T$-$S$-path is a $S$-path that contains vertices of $T$. |
all graphs | e | arbitrary |

[BHJ18a] | directed $S$-$T$-$S$-paths$S.T \subseteq V(G)$. A directed $S$-$T$-$S$-path is a directed $S$-path that contains vertices of $T$. |
all digraphs | v/e | arbitrary |

[BHJ18a] | odd/even directed $S$-paths$S\subseteq V(G)$ and a directed $S$-path is a directed path with both endpoints in $S$. |
all digraphs | v/e | arbitrary |

### Lower bounds for cycles

Ref. | Guest class | Host class | T. | Gap at least | Remark |
---|---|---|---|---|---|

[Tuz90] | triangles | all graphs | e | $\geq 2k$ | |

[Tuz90] | directed triangles | directed graphs without 3 edge-disjoint directed triangles | e | 3 | |

[Lov65] | cycles | graphs without 2 vertex-disjoint cycles | v | =3 | |

[Vos69] | cycles | graphs without 3 vertex-disjoint cycles | v | =6 | |

[Vos67] | cycles | graphs without 4 vertex-disjoint cycles | v | $\geq 9$ | see [Vos69] |

[EP65] | cycles | all graphs | v | $\Omega(k \log k)$ | |

[Sim67] | cycles | all graphs | v | $> \left (\frac{1}{2} + o(1) \right ) k \log k$ | |

[Vos69] | cycles | all graphs | v | $\geq \frac{1}{8} k \log k$ | see [Vos68] |

[KLL02] | cycles | planar graphs | v | $\geq2k$ | |

[MYZ13] | cycles | planar graphs | e | $\geq 4k-c$, $c\in \mathbb{R}$ | |

[DNL87] [Ree99] | odd cycles | all graphs | v | arbitrary | |

[DNL87] | cycles of length $p \mod t$, for every fixed $p \in \{1, \dots, t-1\}$ | all graphs | v | arbitrary | |

[Ree99] | odd cycles | all graphs | e | arbitrary | |

[Tho01] | odd cycles | planar graphs | v | $\geq 2k$ | |

[KV04] | odd cycles | planar graphs | e | $\geq 2k$ | |

[PW12] | $S$-cycles | all graphs | v | $\Omega(k\log k)$ | |

[KKL19] | $S$-cycles | graphs without two disjoint $S$-cycles | v | =4 | |

[KK13] | odd $S$-cycles | all graphs | v | arbitrary | |

[Bru19] | even $S$-cycles of length $\geq \ell$, for every fixed $\ell \geq 5$ | all graphs | e | arbitrary | |

[Bru19] | $S$-cycles of length $0 \mod t$, for every fixed $t > 2$ | all graphs | e | arbitrary | |

[Sey95] | directed cycles | all digraphs | vActually proved for fractional packings | $\frac{1}{30} k \ln k$ | |

[GW96] | directed cycles | planar digraphs | v | $\frac{3}{2}k$ | simpler: a $C_5$ where every edge is replaced by a directed triangle |

[KK12] | directed $S$-cycles | all digraphs | v/e | arbitrary | |

[KKKK13] | odd directed $S$-cycles | all digraphs | v | arbitrary | |

[Bir03] | cycles of length $\geq 4$ | graphs without 2 vertex-disjoint cycles of length $\geq 4$ | v | =4 | |

[Bir03] | cycles of length $\geq 5$ | graphs without 2 vertex-disjoint cycles of length $\geq 5$ | v | =5 | |

[FH14] | cycles of length $\geq t$ | all graphs | v | $\Omega(k\log k)$, $t$ fixed | |

[FH14] | cycles of length $\geq t$ | all graphs | v | $\Omega(t)$, $k$ fixed | |

[MNŠW17] | cycles of length $\geq t$ | all graphs | v | $\geq(k-1)t$ | |

[MNŠW17] | cycles of length $\geq t$ | all graphs | v | $\geq \frac{(k-1)\log k}{8}$ | |

[BHJ16] | cycles of length $\geq t$ | all graphs | e | $O(k \log k + kt)$ | |

[Sim67] | dumb-bellsA dumb-bell is a graph obtained by connecting two cycles by a (non-trivial) path. |
all graphs | v | $>(1+o(1))k \log k$ |

### Lower bounds for minors

Notice that if $H$ is a subcubic graph, then any negative result about the Erdős-Pósa property of $\mathcal{T}(H)$ implies the same for $\mathcal{M}(H)$. Therefore the table below should be completed with the negative results for subcubic $H$’s that are presented in the following section, in particular those of [BHJ18b].

Ref. | Guest class | Host class | T. | Gap at least |
---|---|---|---|---|

[?]This follows from the existence of $n$-vertex 3-regular graphs with treewidth $\Omega(n)$ and girth $\Omega(\log n)$ (Ramanujan graphs, for instance). | $\mathcal{M}(H)$, for every $H$ that has a cycle | all graphs | v/e | $\Omega(k \log k)$ |

[RS86] | $\mathcal{M}(H)$, for every non-planar $H$ | graphs of same Euler genus as $H$ | v | arbitrary |

[RT17] | $\mathcal{M}(H)$, for every non-planar $H$ | graphs of same Euler genus as $H$ | e | arbitrary |

[AKKW16] | $\vec{\mathcal{M}}(H)$, for every $H$ that is not butterfly-minor of the cylindrical grid | all digraphs | v | arbitrary |

[BJS18] | $\mathcal{M}(H)$, for some labelled $H$ that is not 2-connected (and other cases)See the paper for the details and the definition of minors in labelled graphs. | all graphs | v | arbitrary |

[KM19] | graphs of $\mathcal{M}(H)$ meeting $\geq \ell$ of the prescribed subsets, for $\ell \in \mathbb{N}_{\geq 1}$ and $H$ a connected planar graph with $\leq \ell-2$ connected components (detailsThe host graph $G$ comes with a number of prescribed vertex subsets and the guest graphs are those subgraphs of $G$ that can be constracted to $H$ and intersect $\geq \ell$ of the prescribed subsets.) | square grids with prescribed subsets | v | arbitrary |

### Lower bounds for topological minors

Ref. | Guest class | Host class | T. | Gap at least |
---|---|---|---|---|

[Tho88] | $\mathcal{T}(H)$, for every planar $H$ that has no plane embedding with all degree-$(\geq 4)$ vertices on the same face | all graphs | v | arbitrary |

[RT17] | $\mathcal{T}(H)$, for every $H$ non-planar | graphs of same Euler genus as $H$ | v | arbitrary |

[Tho88] | $\mathcal{T}(H)$, for infinitely many trees $H$ with $\Delta(H)=4$ | planar graphs | e | arbitrary |

[RT17] | $\mathcal{T}(H)$, for every $H$ subcubic and non-planar | graphs of same Euler genus as $H$ | e | arbitrary |

[BHJ18b] | $\mathcal{T}(\mathrm{Grid}_{2\times k})$, for every $k\geq 71$ | a class of graphs of treewidth $\leq 8$ | e | arbitrary |

[BHJ18b] | $\mathcal{T}(H)$, for every subcubic tree $H$ of pathwidth $\geq 19$ | a class of graphs of treewidth $\leq 6$ | e | arbitrary |

[AKKW16] | $\vec{\mathcal{T}}(H)$, for every $H$ that is not subdivision of the cylindrical wall | all digraphs | v | arbitrary |

### Lower bounds for immersions

Ref. | Guest class | Host class | T. | Gap at least |
---|---|---|---|---|

copying [Tho88] | $\mathcal{I}(H)$, for infinitely many trees $H$ with $\Delta(H)=4$ | planar graphs | e | arbitrary |

[Liu15] | $\mathcal{I}(H)$ for some $H$Actually this is not proved in the preprint [Liu15] but stated as a consequence of a manuscript of the author that is not availlable online. A description of the graphs $H$ such that $\mathcal{I}(H)$ does not have the edge-EP property in 3-edge-connected graphs is not given in [Liu15]. | 3-edge-connected graphs | e | arbitrary |

[RT17] | $\mathcal{I}(H)$, for every $H$ non-planar | graphs of same Euler genus as $H$ | v | arbitrary |

[RT17] | $\mathcal{I}(H)$, for every $H$ subcubic and non-planar | graphs of same Euler genus as $H$ | e | arbitrary |

[GKRT17] | $\mathcal{I}(H)$, for some 3-connected $H$ with $\Delta(H)=4$ | planar graphs | e | arbitrary |

[KK18a] | $\mathcal{I}(K_5)$ | 3-edge-connected graphs | e | arbitrary |

### Lower bounds for induced patterns

Ref. | Guest class | Host class | T. | Gap at least |
---|---|---|---|---|

[KK18b] | cycles of length $\geq t$, for every $t \geq 5$ | all graphs | v | arbitrary |

[KR18] | $\mathcal{T}(K_{n,m})$, for every $n\geq 2$ and $m \geq 3$ | all graphs | v | arbitrary |

[KR18] | $\mathcal{T}(F)$, for every $F$ forest where two $(\geq 3)$-degree vertices lie in the same component | all graphs | v | arbitrary |

[KR18] | $\mathcal{T}(H)$, for every $H$ that has an induced cycle of length $\geq 5$ | all graphs | v | arbitrary |

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Last updated: July 2022.