Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult

We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem

How I got to like graph polynomials

For Boris Zilber on his 75th birthday. I trace the roots of my collaboration with Boris Zilber, which combines categoricity theory, finite model theory, algorithmics, and combinatorics.Comment: 11 page

Connection Matrices and the Definability of Graph Parameters

In this paper we extend and prove in detail the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers

Planar graph coloring avoiding monochromatic subgraphs : trees and paths make it difficult

We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic

Polar permutation graphs are polynomial-time recognisable *

Abstract Polar graphs generalise bipartite graphs, cobipartite graphs, and split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here, we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs

Efficient domination and polarity

The thesis considers the following graph problems: Efficient (Edge) Domination seeks for an independent vertex (edge) subset D such that all other vertices (edges) have exactly one neighbor in D. Polarity asks for a vertex subset that induces a complete multipartite graph and that contains a vertex of every induced P_3. Monopolarity is the special case of Polarity where the wanted vertex subset has to be independent. These problems are NP-complete in general, but efficiently solvable on various graph classes. The thesis sharpens known NP-completeness results and presents new solvable cases

New results on generalized graph colorings

For graph classes P1,..., Pk, Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V1,...,Vk so that Vj induces a graph in the class Pj (j = 1,2,...,k). If P1 = · · · = Pk is the class of edgeless graphs, then this problem coincides with the standard vertex k-colorability, which is known to be NP-complete for any k ≥ 3. Recently, this result has been generalized by showing that if all Pi’s are additive induced-hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs. Clearly, a similar result follows when all the Pi’s are co-additive. In this paper, we study the problem where we have a mixture of additive and co-additive classes, presenting several new results dealing both with NPhard and polynomial-time solvable instances of the problem