Finding large H-colorable subgraphs in hereditary graph classes
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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 {P_5,F}-free graphs in polynomial time, whenever F is a threshold graph; ∙ in {P_5,bull}-free graphs in polynomial time; ∙ in P_5-free graphs in time n^𝒪(ω(G)); ∙ in {P_6,1-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, combining the mentioned algorithms for P_5-free and for {P_6,1-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 P_5-free graphs, if we allow loops on H.
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