Solving problems on generalized convex graphs via mim-width

BRETTELL, NICK; BONOMO, FLAVIA; PAULUSMA, DANIËL; MUNARO, ANDREA

Halifax

Simposio; 17th Algorithm and Data Structures Symposium; 2021

A bipartite graph $G=(A,B,E)$ is ${\cal H}$-convex, for some family of graphs ${\cal H}$, if there exists a graph $H\in {\cal H}$ with $V(H)=A$ such that the set of neighbours in $A$ of each $b\in B$ induces a connected subgraph of~$H$. Many $\mathsf{NP}$-complete problemsbecome polynomial-time solvable for ${\mathcal H}$-convex graphs when ${\mathcal H}$ is the set of paths. In this case, the class of ${\mathcal H}$-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 ${\mathcal H}$-convex graphs where (i) ${\mathcal H}$ is the set of cycles, or (ii) ${\mathcal H}$ 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 ${\mathcal H}$-convex graphs is unbounded if ${\mathcal H}$ 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 ${\cal H}$-convex graphs.