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 nO(ω(G)); and in {P6,1−subdividedclaw}-free graphs in time nO(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−subdividedclaw}-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.