We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Although our algorithm is simple and intuitive, its correctness proof relies on a series of non-trivial structural results, and turns out to be surprisingly complex. Threshold graphs are a well-studied subclass of interval graphs and of spl...

In this paper, we provide a graph theoretical framework that allows us to formally define formations of multiple vehicles and the issues arising in graph realizations and unicity and their connections to stability of formations. The notion of graph rigidity is crucial in identifying the shape variables of a formation. This eventually leads to tools for formation stabilization, tacking, and form...

Let , be a graph with vertex set and edge set . Let then is said to be a local metric basis of , if for any two adjacent vertices , ⁄ , there exists a such that , , . The minimum cardinality of local metric basis is called the local metric dimension (lmd) of graph G. In this paper we investigate the local metric basis and local metric dimension of Cyclic Split Graph .

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

This paper deals with the graph isomorphism (GI) problem for two graph classes: chordal bipartite graphs and strongly chordal graphs. It is known that GI problem is GI complete even for some special graph classes including regular graphs, bipartite graphs, chordal graphs, comparability graphs, split graphs, and k-trees with unbounded k. On the other side, the relative complexity of the GI probl...

A graph is called a cograph contraction if it is obtained from a cograph (a graph with no induced path on four vertices) by contracting some pairwise disjoint independent sets and then making the ‘‘contracted’’ vertices pairwise adjacent. Cograph contractions are perfect and generalize cographs and split graphs. This article gives a good characterization of cograph contractions, solving a probl...

We show that deciding whether a given graph G of size m has a unique perfect matching as well as finding that matching, if it exists, can be done in time O(m) if G is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.

This paper presents a phenomenon which sometimes occurs in tetravalent bipartite locally dart-transitive graphs, called Base Graph–Connection Graph dissection. In this dissection, each white vertex is split into two vertices of valence 2 so that the connected components result are isomorphic. Given whose subdivision isomorphic to component, and Connection Graph, describes how overlap, we can, s...

A parallel knock-out scheme for a graph proceeds in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is KO-reducible if there exists such a scheme that eliminates every vertex in the graph. The Parallel Knock-Out problem is to decide whether a graph G is KO-reducible. This problem is known to be NP-complete and has been studied for several graph ...

We define the A4-structure of a graph G to be the 4-uniform hypergraph on the vertex set of G whose edges are the vertex subsets inducing 2K2, C4, or P4. We show that perfection of a graph is determined by its A4-structure. We relate the A4structure to the canonical decomposition of a graph as defined by Tyshkevich [Discrete Mathematics 220 (2000) 201-238]; for example, a graph is indecomposabl...