Computing Homomorphisms in Hereditary Graph Classes: The Peculiar Case of the 5-Wheel and Graphs with No Long Claws

For graphs G and H, an H-coloring of G is an edge-preserving mapping from V(G) to V(H). In the H-Coloring problem the graph H is fixed and we ask whether an instance graph G admits an H-coloring. A generalization of this problem is H-ColoringExt, where some vertices of G are already mapped to vertices of H and we ask if this partial mapping can be extended to an H-coloring. We study the complexity of variants of H-Coloring in F-free graphs, i.e., graphs excluding a fixed graph F as an induced subgraph. For integers a,b,c ? 1, by S_{a,b,c} we denote the graph obtained by identifying one endvertex of three paths on a+1, b+1, and c+1 vertices, respectively. For odd k ? 5, by W_k we denote the graph obtained from the k-cycle by adding a universal vertex. As our main algorithmic result we show that W_5-ColoringExt is polynomial-time solvable in S_{2,1,1}-free graphs. This result exhibits an interesting non-monotonicity of H-ColoringExt with respect to taking induced subgraphs of H. Indeed, W_5 contains a triangle, and K_3-Coloring, i.e., classical 3-coloring, is NP-hard already in claw-free (i.e., S_{1,1,1}-free) graphs. Our algorithm is based on two main observations: 1) W_5-ColoringExt in S_{2,1,1}-free graphs can be in polynomial time reduced to a variant of the problem of finding an independent set intersecting all triangles, and 2) the latter problem can be solved in polynomial time in S_{2,1,1}-free graphs. We complement this algorithmic result with several negative ones. In particular, we show that W_5-Coloring is NP-hard in P_t-free graphs for some constant t and W_5-ColoringExt is NP-hard in S_{3,3,3}-free graphs of bounded degree. This is again uncommon, as usually problems that are NP-hard in S_{a,b,c}-free graphs for some constant a,b,c are already hard in claw-free graph