We present efficient parallel algorithms for recognizing chordal graphs and locating all maximal cliques of a chordal graph G=(V,E). Our techniques are based on partitioning the vertex set V using information contained in the distance matrix of the graph. We use these properties to formulate parallel algorithms which, given a graph G=(V,E) and its adjacency-level sets, decide whether or not G i...

We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The edge constraints are given as a pair of graphs one of which contains the other and one of which is chordal, and the objects we look at are the chordal graphs contained in one and containing the other. This setting is a natural generalization of chordal completions and deletions. For the listing prob...

Given a graph G = (V, E) and a set S ⊆ V , a set U ⊆ V is a subset feedback vertex set of (G, S ) if no cycle in G[V \ U] contains a vertex of S . The Subset Feedback Vertex Set problem takes as input G, S , and an integer k, and the question is whether (G, S ) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are ...

We prove that every planar graph on n vertices is contained in a chordal graph with at most cn log n edges for some absolute constant c and this is best possible to within a constant factor.

Let G be a simple undirected graph on n vertices, and let I(G) ⊆ R = k[x 1 ,. .. , x n ] 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 im...

The objective of this paper is to advance the view that solving the all-pairs shortest path (APSP) problem for a chordal graph G is a two-step process: the first step is determining vertex pairs at distance two (i.c., computing C’) and the second step is finding the vcrtcx pairs at distance three or more. The main technical result here is that the APSP problem for a chordal graph can be solved ...

Vertex deletion problems are at the heart of parameterized complexity. For a graph class F , the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the resulting graph belongs to F . Whether Perfect Deletion is fixed-parameter tractable, and whether Chordal Deletion admits a polynomial kernel, when p...

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 treewi...

A fundamental problem in computational biology is the phylogeny reconstruction for a set of specific organisms. One of the graph theoretical approaches is to construct a similarity graph on the set of organisms where adjacency indicates evolutionary closeness, and then to reconstruct a phylogeny by computing a tree interconnecting the organisms such that leaves in the tree are labeled by the or...