Finding Large $H$-Colorable Subgraphs in Hereditary Graph Classes

Abstract

We study the Max Partial H-Coloring problem: given a graph G, find the largest induced subgraph of G that admits a homomorphism into H, where H is a fixed pattern graph without loops. Note that when H is a complete graph on k vertices, the problem reduces to finding the largest induced k-colorable subgraph, which for k = 2 is equivalent (by complementation) to Odd Cycle Transversal.We prove that for every fixed pattern graph H without loops, Max Partial H-Coloring can be solved: in {P5, F }-free graphs in polynomial time, whenever F is a threshold graph; in {P5, bull}-free graphs in polynomial time; in P5-free graphs in time n O(ω(G)) ; in {P6, 1-subdivided claw}-free graphs in time n O(ω(G) 3 ) . Here, n is the number of vertices of the input graph G and ω(G) is the maximum size of a clique in G. Furthermore, by combining the mentioned algorithms for P5-free and for {P6, 1-subdivided claw}-free graphs with a simple branching procedure, we obtain subexponential-time algorithms for Max Partial H-Coloring in these classes of graphs.Finally, we show that even a restricted variant of Max Partial H-Coloring is NP-hard in the considered subclasses of P5-free graphs, if we allow loops on H.

Complexity of the list homomorphism problem in hereditary graph classes

“…Let us point out that all mentioned hardness proofs rule out the existence of subexponential-time algorithms, unless the ETH fails. Furthermore, all algorithmic results hold even for L k C , except for the case (k, t) = (4,6), which is NP-complete in the list setting [20].…”

“…First, Chudnovsky et al [3] showed that for k ∈ {5, 7, 9}∪ [10; ∞), the LH (C k ) problem can be solved in polynomial time for P 9 -free graphs. Very recently, Chudnovsky et al [4] studied some further generalization of the homomomorphism problem in subclasses of P 6 -free graphs. Furthermore, the already mentioned 2 O( √ n log n) -time algorithm by Groenland et al [21] actually works for LH (H) for a large family of graphs H: the requirement is that H does not contain two vertices with two common neighbors.…”

List k-colouring P-free graphs: A Mim-width perspective

“…Theorem 6 has other applications as well. For a graph G, let ω(G) denote the size of a maximum clique in G. Chudnovsky et al [9] gave for the class of (K 1 1,3 , P 6 )-free graphs, an n O(ω(G) 3 ) -time algorithm for Max Partial H-Colouring, a problem equivalent to Independent Set if H = P 1 and to Odd Cycle Transversal if H = P 2 . In other words, Max Partial H-Colouring is polynomial-time solvable for (K 1 1,3 , P 6 )-free graphs with bounded clique number.…”

“…In other words, Max Partial H-Colouring is polynomial-time solvable for (K 1 1,3 , P 6 )-free graphs with bounded clique number. Chudnovsky et al [9] noted that Max Partial H-Colouring is polynomial-time solvable for graph classes whose mim-width is bounded and quickly computable. Hence, Theorem 6 generalizes their result for Max Partial H-Colouring to (K 1 1,s , P t )-free graphs with bounded clique number, for any s ≥ 1 and t ≥ 1.…”

“…Hence, Theorem 6 generalizes their result for Max Partial H-Colouring to (K 1 1,s , P t )-free graphs with bounded clique number, for any s ≥ 1 and t ≥ 1. However, the running time of the corresponding algorithm is worse than n O(ω(G) 3 ) (see [9] for details).…”