[Paper Review] On Distance-$d$ Independent Set and other problems in graphs with few minimal separators

Pedro Montealegre, Ioan Todinca|arXiv (Cornell University)|Jul 15, 2016

Advanced Graph Theory Research7 references4 citations

TL;DR

This paper extends Fomin and Villanger's polynomial-time algorithms for problems on graphs with few minimal separators by showing that Distance-$d$ Independent Set is solvable in polynomial time on graphs with polynomially many minimal separators ($\mathcal{G}_{\operatorname{poly}}$) for any even $d$. It proves that the $(d-1)$-th power of a graph has no more minimal separators than the original graph when $d-1$ is odd, enabling dynamic programming over minimal separators for even $d$. The result also extends to Connected Vertex Cover and Connected Feedback Vertex Set on subclasses like chordal and circular-arc graphs.

ABSTRACT

Fomin and Villanger (STACS 2010) proved that Maximum Independent Set, Feedback Vertex Set, and more generally the problem of finding a maximum induced subgraph of treewith at most a constant $t$, can be solved in polynomial time on graph classes with polynomially many minimal separators. We extend these results in two directions. Let $\Gpoly$ be the class of graphs with at most $\poly(n)$ minimal separators, for some polynomial $\poly$. We show that the odd powers of a graph $G$ have at most as many minimal separators as $G$. Consequently, extsc{Distance-$d$ Independent Set}, which consists in finding maximum set of vertices at pairwise distance at least $d$, is polynomial on $\Gpoly$, for any even $d$. The problem is NP-hard on chordal graphs for any odd $d \geq 3$. We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of $\Gpoly$ including chordal and circular-arc graphs, and we discuss variants of independent domination problems.

Motivation & Objective

• To extend polynomial-time solvability of graph problems beyond treewidth-bounded induced subgraphs to problems with distance and connectivity constraints.
• To resolve the complexity of Distance-$d$ Independent Set on $\mathcal{G}_{\operatorname{poly}}$ for even and odd $d$, particularly in chordal and weakly chordal graphs.
• To investigate whether Connected Vertex Cover and Connected Feedback Vertex Set are polynomial-time solvable on $\mathcal{G}_{\operatorname{poly}}$ subclasses like chordal and circular-arc graphs.
• To explore the interplay between minimal separators, graph powers, and CMSO-expressible properties in dynamic programming frameworks.

Proposed method

• Leveraging the fact that the $(d-1)$-th power of a graph $G$ has at most as many minimal separators as $G$ when $d-1$ is odd, enabling reduction of Distance-$d$ Independent Set to problems on $G^{d-1}$.
• Applying dynamic programming over minimal separators and potential maximal cliques, as in Fomin and Villanger (2010), to solve Distance-$d$ Independent Set on $\mathcal{G}_{\operatorname{poly}}$ for even $d$.
• Using reductions from Connected Vertex Cover to Red-Blue Dominating Set on bipartite graphs in $\mathcal{G}_{\operatorname{poly}}$, showing polynomial solvability when Red-Blue Dominating Set is polynomial.
• Establishing that Red-Blue Dominating Set is polynomial on chordal bipartite and circle bipartite graphs via known results on tree augmentation and fundamental graphs.
• Extending the framework of Fomin et al. (2013) to include connectivity and domination constraints via CMSO logic and induced subgraph conditions.
• Proving that the number of minimal separators is preserved or increased by at most $\mathcal{O}(n)$ under key reductions, preserving polynomial-time solvability.

Experimental results

Research questions

• RQ1 Is Distance-$d$ Independent Set polynomial-time solvable on $\mathcal{G}_{\operatorname{poly}}$ for even $d$?
• RQ2 Can Connected Vertex Cover be solved in polynomial time on chordal and circular-arc graphs, which are subclasses of $\mathcal{G}_{\operatorname{poly}}$?
• RQ3 Does the number of minimal separators remain bounded under graph powers, particularly for odd powers?
• RQ4 Is Red-Blue Dominating Set polynomial on bipartite graphs in $\mathcal{G}_{\operatorname{poly}}$, such as chordal bipartite and circle bipartite graphs?
• RQ5 Can the dynamic programming framework of Fomin and Villanger be extended to handle connectivity and domination constraints in induced subgraphs?

Key findings

• Distance-$d$ Independent Set is solvable in polynomial time on $\mathcal{G}_{\operatorname{poly}}$ for any even $d$, due to the preservation of minimal separator count under odd powers of the graph.
• For any odd $d \geq 3$, the problem is NP-hard on $\mathcal{G}_{\operatorname{poly}}$ when the polynomial bound on minimal separators is asymptotically larger than $n$.
• Connected Vertex Cover is polynomial-time solvable on chordal and circular-arc graphs, as these are subclasses of $\mathcal{G}_{\operatorname{poly}}$ and Red-Blue Dominating Set is polynomial on their bipartite counterparts.
• Red-Blue Dominating Set is polynomial on chordal bipartite graphs, as shown by Kanj et al., and on circle bipartite graphs via a reduction to the Tree Augmentation problem on planar multigraphs.
• The number of minimal separators increases by at most $\mathcal{O}(n)$ under the reduction from Connected Vertex Cover to Red-Blue Dominating Set, preserving polynomial-time solvability on $\mathcal{G}_{\operatorname{poly}}$.
• The framework extends to problems requiring induced subgraphs of bounded treewidth with additional CMSO constraints, such as connectivity or domination, on $\mathcal{G}_{\operatorname{poly}}$ subclasses.