The directed vertex leafage of a chordal graph G is the smallest integer k such that G is the intersection graph of subtrees of a rooted directed tree where each subtree has at most k leaves. In this note, we show how to find in time O(kn) an optimal colouring, a maximum independent set, a maximum clique, and an optimal clique cover of an n-vertex chordal graph G with directed vertex leafage k ...

Using the characteristic property of chordal graphs that they are the intersection graphs of subtrees of a tree, Erich Prisner showed that every chordal graph admits an eccentricity 2approximating spanning tree. That is, every chordal graph G has a spanning tree T such that eccT (v) − eccG(v) ≤ 2 for every vertex v, where eccG(v) (eccT (v)) is the eccentricity of a vertex v in G (in T , respect...

Weighted independent domination is an NP-hard graph problem, which remains computationally intractable in many restricted graph classes. In particular, the problem is NP-hard in the classes of sat-graphs and chordal graphs. We strengthen these results by showing that the problem is NP-hard in a proper subclass of the intersection of sat-graphs and chordal graphs. On the other hand, we identify ...

Hadwiger’s conjecture states that for every graph G, χ(G) ≤ η(G), where χ(G) is the chromatic number and η(G) is the size of the largest clique minor in G. In this work, we show that to prove Hadwiger’s conjecture in general, it is sufficient to prove Hadwiger’s conjecture for the class of graphs F defined as follows: F is the set of all graphs that can be expressed as the square graph of a spl...

In this paper, we present a linear-time algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a linear-time algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n) to O(n+m). We also devise a simple linear-time algorithm for interval graph recognition where no complic...

A graph G is chordal-bipartite probe if its vertices can be partitioned into two sets P (probes) and N (non-probes) where N is a stable set and such thatG can be extended to a chordal-bipartite graph by adding edges between non-probes. A bipartite graph is called chordal-bipartite if it contains no chordless cycle of length strictly greater than 5. Such probe/non-probe completion problems have ...

This paper studies routing vulnerability in networks modeled by chordal ring graphs. In a chordal ring graph, the vertices are labeled in Z2n and each even vertex i is adjacent to the vertices i + a, i + b, i + c, where a, b, and c are different odd integers. Our study is based on a geometrical representation that associates to the graph a tile which periodically tessellates the plane. Using th...

A graph is chordal if it does not contain any induced cycle of size greater than three. An alternative characterization of chordal graphs is via a perfect elimination ordering, which is an ordering of the vertices such that, for each vertex v, the neighbors of v that occur later than v in the ordering form a clique. Akcoglu et al [2] define an extension of chordal graphs whereby the neighbors o...

In a graph G, a vertex is said to dominate itself and all of its neighbors. For a fixed positive integer k, the k-tuple domination problem is to find a minimum sized vertex subset in a graph such that every vertex in the graph is dominated by at least k vertices in this set. The current paper studies k-tuple domination in graphs from an algorithmic point of view. In particular, we give a linear...

We improve the current complexities for maintaining a chordal graph by starting with an empty graph and repeatedly adding or deleting edges.