Enumerating minimal vertex covers and dominating sets with capacity and/or connectivity constraints

TL;DR

It is shown that the minimal connected vertex cover, minimal connected dominating set, and minimal capacitated vertex cover enumeration problems in 2-degenerated bipartite graphs are at least as hard as enumerating minimal transversals in hypergraphs.

Abstract

In this paper, we consider the problems of enumerating minimal vertex covers and minimal dominating sets with capacity and/or connectivity constraints. We develop polynomial-delay enumeration algorithms for these problems on bounded-degree graphs. For the case of minimal connected vertex covers, our algorithms run in polynomial delay even on the class of $d$-claw free graphs, extending the result on bounded-degree graphs, and in output quasi-polynomial time on general graphs. To complement these algorithmic results, we show that the problems of enumerating minimal connected vertex covers, minimal connected dominating sets, and minimal capacitated vertex covers in $2$-degenerated bipartite graphs are at least as hard as enumerating minimal transversals in hypergraphs.

Figures (3)

• Figure 1: The left figure illustrates the constructed graph $G$. Filled circles indicate a minimal connected vertex cover of $G$. The right figure illustrates a part of the construction of a $2$-degenerate graph $G'$ from $G$.
• Figure 2: The figure depicts matchings $M_X$ (black edges) and $M_Y$ (red edges) in $H$.
• Figure 3: The left figure illustrates the constructed graph $G$. Filled circles indicate a minimal capacitated vertex cover of $(G, c)$ and arrows indicate the function $\alpha$. The right figure illustrates a part of the construction of a $2$-degenerate graph $G'$ from $G$.

Theorems & Definitions (38)

• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1: e.g., KobayashiKW:arXiv:Efficient:2020CohenKS:JCSS:Generating:2008ConteGMUV:SICOMP:Proximity:2022
• Lemma 1
• proof
• Theorem 2
• Theorem 3
• Theorem 4