Solving Problems on Generalized Convex Graphs via Mim-Width

A bipartite graph G=(A,B,E)G=(A,B,E) is HH-convex, for some family of graphs HH, if there exists a graph H∈HH∈H with V(H)=AV(H)=A such that the set of neighbours in A of each b∈Bb∈B induces a connected subgraph of H. Many NPNP-complete problems become polynomial-time solvable for HH-convex graphs when HH is the set of paths. In this case, the class of HH-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of HH-convex graphs where (i) HH is the set of cycles, or (ii) HH 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 HH-convex graphs is unbounded if HH is the set of trees with arbitrarily large maximum degree or an 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. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of HH-convex graphs.