Graph-Theoretic Concepts in Computer Science

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 $c(u)\neq c(v)$ 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.

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

The first issue of the "Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg" was published in the year 1921. This international mathematical journal has since then provided a forum for significant research contributions. The journal covers all central areas of pure mathematics, such as algebra, complex analysis and geometry, differential geometry and global analysis, graph theory and discrete mathematics, Lie theory, number theory, and algebraic topology.

Abhandlungen aus dem mathematischen seminar der Universität Hamburg

A graph is H -free if it does not contain an induced subgraph isomorphic to H . We denote by P t and C t the path and the cycle on t vertices, respectively. In this paper, we prove that 4-COLORING is NP-complete for P 7 -free graphs, and that 5-COLORING is NP-complete for P 6 -free graphs. The second result is the first NP-hardness result for k -COLORING P 6 -free graphs. These two results improve all previous NP-complete results on k -coloring P t -free graphs, and almost complete the classification of complexity of k -COLORING P t -free graphs for k ? 4 and t ? 1 , leaving as the only missing case 4-COLORING of P 6 -free graphs. We expect that 4-COLORING is polynomial time solvable for P 6 -free graphs; in support of this, we describe a polynomial time algorithm for 4-COLORING P 6 -free graphs which are also banner-free, where banner is the graph obtained from C 4 by adding a new vertex and making it adjacent to exactly one vertex on the C 4 .

Improved complexity results on k -coloring P t -free graphs

In this paper we present a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002 Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1,2,3}, and gives an explicit coloring if one exists

/pdf/three-coloring-and-list-three-coloring-of-graphs-without-53r0sy5oqb.pdf

Three-coloring and list three-coloring of graphs without induced paths on seven vertices

http://www.cs.toronto.edu/~stacho/public/mpt-dam.pdf

Max Point-Tolerance Graphs

Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.

/pdf/contact-representations-of-planar-graphs-extending-a-partial-narmd573qp.pdf

Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

http://www.cs.toronto.edu/~stacho/public/polar-final.pdf

Polarity of chordal graphs

We investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. We propose a novel decomposition of reduced clique graphs based on two operations: edge contraction and removal of the edges of a split. Based on this decomposition, we characterize asteroidal sets in chordal graphs, and discuss chordal graphs that admit a tree representation with a small number of leaves.

/pdf/reduced-clique-graphs-of-chordal-graphs-4phgkagq8q.pdf

Reduced clique graphs of chordal graphs

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, known as B0 -VPG graphs. Recognizing these graphs is an NP-hard problem. In light of this, we focus on their subclasses. In the paper, we describe polynomial time algorithms for recognizing chordal B0 -VPG graphs, and for recognizing B0 -VPG graphs that have a representation on a grid with 2 rows.

http://www.cs.toronto.edu/~stacho/public/b0vpg3.pdf

Juraj Stacho

Papers

Max Point-Tolerance Graphs

Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

Polarity of chordal graphs

Reduced clique graphs of chordal graphs

Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid