Results on independent sets in categorical products of graphs, the ultimate categorical independence ratio and the ultimate categorical independent domination ratio

TL;DR

This paper investigates the computational complexity and algorithms for maximum independent sets in categorical graph products, introducing polynomial-time algorithms for specific graph classes and analyzing the ultimate ratios of independence and domination.

Contribution

It provides polynomial-time algorithms for maximum independent sets in categorical products of certain graph classes and studies the ultimate ratios of independence and domination in various graphs.

Findings

Polynomial-time algorithms for cographs and splitgraphs.

NP-completeness results for planar graphs with K_4.

A PTAS for planar graphs' ultimate independence ratio.

Abstract

We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a graph G is defined as lim_{k --> infty} \alpha(G^k)/n^k. The ultimate categorical independence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. When G is a planar graph of maximal degree three then alpha(G \times K_4) is NP-complete. We present a PTAS for the ultimate categorical independence ratio of planar graphs. We present an O^*(n^{n/3}) exact, exponential algorithm for general graphs. We prove that the ultimate categorical independent domination ratio for complete multipartite graphs is zero, except when the graph is complete bipartite with color classes of equal size (in which case it is 1/2).