Brian Moore

Bio: Brian Moore is an academic researcher from Wilfrid Laurier University. The author has contributed to research in topics: 1-planar graph & Partial k-tree. The author has an hindex of 2, co-authored 2 publications receiving 29 citations.

A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs

Abstract: For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.

A Survey on the Computational Complexity of Colouring Graphs with Forbidden Subgraphs

Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u) eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.

Exhaustive generation of k-critical H-free graphs

Abstract: We describe an algorithm for generating all k-critical H-free graphs, based on a method of Hoang et al. (A graph G is k-critical H-free if G is H-free, k-chromatic, and every H-free proper subgraph of G is (k−1)-colorable). Using this algorithm, we prove that there are only finitely many 4-critical (P7,Ck)-free graphs, for both k=4 and k=5. We also show that there are only finitely many 4-critical (P8,C4)-free graphs. For each of these cases we also give the complete lists of critical graphs and vertex-critical graphs. These results generalize previous work by Hell and Huang, and yield certifying algorithms for the 3-colorability problem in the respective classes. In addition, we prove a number of characterizations for 4-critical H-free graphs when H is disconnected. Moreover, we prove that for every t, the class of 4-critical planar Pt-free graphs is finite. We also determine all 52 4-critical planar P7-free graphs. We also prove that every P11-free graph of girth at least five is 3-colorable, and show that this is best possible by determining the smallest 4-chromatic P12-free graph of girth at least five. Moreover, we show that every P14-free graph of girth at least six and every P17-free graph of girth at least seven is 3-colorable. This strengthens results of Golovach et al.

Complexity of Coloring Graphs without Paths and Cycles

Abstract: Let P t and C l denote a path on t vertices and a cycle on l vertices, respectively. In this paper we study the k-COLORING problem for (P t ,C l)-free graphs. It has been shown by Golovach, Paulusma, and Song that when l = 4 all these problems can be solved in polynomial time. By contrast, we show that in most other cases the k-COLORING problem for (P t ,C l)-free graphs is NP-complete. Specifically, for l = 5 we show that k-COLORING is NP-complete for (P t ,C 5)-free graphs when k ≥ 4 and t ≥ 7; for l ≥ 6 we show that k-COLORING is NP-complete for (P t ,C l)-free graphs when k ≥ 5, t ≥ 6; and additionally, we prove that 4-COLORING is NP-complete for (P t ,C l)-free graphs when t ≥ 7 and l ≥ 6 with l ≠ 7, and that 4-COLORING is NP-complete for (P t ,C l)-free graphs when t ≥ 9 and l ≥ 6 with l ≠ 9. It is known that, generally speaking, for large k the k-COLORING problem tends to remain NP-complete when one forbids an induced path P t with large t. Our findings mean that forbidding an additional induced cycle C l (with l > 4) does not help. We also revisit the problem of k-COLORING (P t ,C 4)-free graphs, in the case t = 6. (For t = 5 the k-COLORING problem is known to be polynomial even on just P 5-free graphs, for every k.) The algorithms of Golovach, Paulusma, and Song are not practical as they depend on Ramsey-type results, and end up using tree-decompositions with very high widths. We develop more practical algorithms for 3-COLORING and 4-COLORING on (P 6,C 4)-free graphs. Our algorithms run in linear time if a clique cutset decomposition of the input graph is given. Moreover, our algorithms are certifying algorithms. We provide a finite list of all minimal non-k-colorable (P 6,C 4)-free graphs, for k = 3 and k = 4. Our algorithms output one of these minimal obstructions whenever a k-coloring is not found. In fact, we prove that there are only finitely many minimal non-k-colorable (P 6,C 4)-free graphs for any fixed k; however, we do not have the explicit lists for higher k, and thus no certifying algorithms. (We note there are infinitely many non-k-colorable P 5-free, and hence P 6-free, graphs for any given k ≥ 4, according to a result of Hoang, Moore, Recoskie, Sawada, and Vatshelle.)