Let G = (V, E) be an undirected graph and r be a vertex weight function with positive integer values. A subset (clique) D ~_ V is an r-dominat ing set (clique) in G ifffor every vertex v e V there is a vertex u e D with dist(u, v) <~ r(v). This paper contains the following results: (i) We give a simple necessary and sufficient condition for the existence of r-dominating cliques in the case of H...

A path in an edge-colored graph is rainbow if no two edges of it are colored the same. The graph is said to be rainbow connected if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is strong rainbow connected. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or stro...

The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. We prove in this paper that the PATHWIDTH problem is NP-hard for particular subclasses of chordal graphs, and we deduce that the problem remains hard for weighted trees. We also discuss subclasses of chordal graphs for which the problem is polynomial.

We consider the problem of aligned coloring of interval and chordal graphs. These problems have substantial applications to register allocation in compilers and have recently been proven NP-Hard. We provide the first constant approximations: a 4 3 -approximation for interval graphs and a 3 2 -approximation for chordal graphs. We extend our techniques to the problem of minimizing spillage in the...

In this note, we look at which edges must always be added to a given graph G = (V, E), when we want to make it a chordal graph with maximum clique size at most k by adding edges. lThis problem has a strong relation to the (algorithmic) theory of the treewidth of graphs. If {x, y} is an edge in every chordal supergraph of G with maximum clique size at most k, we call the pair necessary for treew...

In an earlier paper we gave efficient algorithms for partitioning chordal graphs into k independent sets and l cliques. This is a natural generalization of the problem of recognizing split graphs, and is NPcomplete for graphs in general, unless k ≤ 2 and l ≤ 2. (Split graphs have k = l = 1.) In this paper we expand our focus and consider generalM -partitions, also known as trigraph homomorphism...

** This thesis studies graph clustering on bipartite and chordal graphs. It describes two types of clustering problems: a partition of the vertex of a graph with minimum diameters of subgraphs; maximizing possible number of edges given fixed number of vertices.

A graph H is a square root of a graph G if two vertices are adjacent in G if and only if they are at distance one or two in H . Computing a square root of a given graph is NP-hard, even when the input graph is restricted to be chordal. In this paper, we show that computing a square root can be done in linear time for a well-known subclass of chordal graphs, the class of trivially perfect graphs...

Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x1, . . . , xn] denote its associated edge ideal. We show that all chordal graphs G are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of I(G) is componentwise linear. Our result complements Faridi’s theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and impl...

mim-width is a recent graph width measure that has seen applications in graph algorithms and problems related to propositional satisfiability. In this paper, we show linear lower bounds for the mim-width of strongly chordal split graphs, co-comparability graphs and circle graphs. This improves and refines lower bounds that were known before, some of them only conditionally. In the case of stron...