Solving problems on generalized convex graphs via mim-width
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A bipartite graph G=(A,B,E) is H-convex, for some family of graphs H, if there exists a graph H∈ H with V(H)=A such that the set of neighbours in A of each b∈ B induces a connected subgraph of H. A variety of well-known 𝖭𝖯-complete problems, including Dominating Set, Feedback Vertex Set, Induced Matching and List k-Colouring, become polynomial-time solvable for ℋ-convex graphs when ℋ is the set of paths. In this case, the class of ℋ-convex graphs is known as the class of convex graphs. The underlying reason is that the class of convex graphs has bounded mim-width. We extend the latter result to families of ℋ-convex graphs where (i) ℋ is the set of cycles, or (ii) ℋ is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can re-prove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of ℋ-convex graphs is unbounded if ℋ is the set of trees with arbitrarily large maximum degree or arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems.
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