The notion of Asteroidal triples was introduced by Lekkerkerker and Boland [6]. D.G.Corneil and others [2], Ekkehard Kohler [3] further investigated asteroidal triples. Walter generalized the concept of asteroidal triples to asteroidal sets [8]. Further study was carried out by Haiko Muller [4]. In this paper we find asteroidal numbers for Direct product of cycles, Direct product of path and cy...

Asteroidal Triple-free (AT-free) graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighbourhood of any vertex in the set leaves the remaining vertices of the set in the same connected...

In this paper we introduce the class of graphs whose complements are asteroidal (star-like) graphs and derive closed formulas for the number of spanning trees of its members. The proposed results extend previous results for the classes of the multi-star and multi-complete/star graphs. Additionally, we prove maximization theorems that enable us to characterize the graphs whose complements are as...

A vertex colouring assigns to each vertex of a graph a colour such that adjacent vertices have different colours. The algorithmic complexity of the Colouring problem, asking for the smallest number of colours needed to vertex-colour a given graph, is known for a large number of graph classes. Notably it is NP-complete in general, but polynomial time solvable for perfect graphs. A triple of vert...

We present O(n’R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs tha...

We present an efficient algorithm for computing the vertex ranking number of an asteroidal triple-free graph. lts running time is bounded by a polynomial in the number of vertices and the number of minimal separators of the input graph.

An asteroidal triple is an independent set of vertices such that each pair is joined by a path that avoids the neighborhood of the third, and a moplex is an extension to an arbitrary graph of a simplicial vertex in a triangulated graph. The main result of this paper is that the investigation of the set of moplexes of a graph is sufficient to conclude as to its having an asteroidal triple. Speci...

An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it contains no asteroidal triple. In this paper, we prove a...

We define the stable degree s(G) of a graph G by s(G) = minU maxv∈U dG(v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k ≥ 3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal nu...