‎Unmixed bipartite graphs have been characterized by Ravindra and‎ ‎Villarreal independently‎. ‎Our aim in this paper is to‎ ‎characterize unmixed $r$-partite graphs under a certain condition‎, ‎which is a generalization of Villarreal's theorem on bipartite‎ ‎graphs‎. ‎Also, we give some examples and counterexamples in relevance to this subject‎.

A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size α = α(G), or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement G are both well-co...

A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size α = α(G), or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement G are both well-co...

For some given graph H, a G is called H-covered if each vertex in contained copy of H. In this note, we determine the maximum number independent sets size t ≥ 3 N-vertex K n -covered graphs and classify extremal graphs. The result answers question proposed by Chakraborti Loh. proof uses an edge-switching operation on hypergraphs which never increases sets.

This paper presents a graph-based formalism for an Ontology Based Access Control (OBAC) system applied to Digital Library (DL) ontology. It uses graph transformations, a graphical specification technique based on a generalization of classical string grammars to nonlinear structures. The proposed formalism provides an executable specification that exploits existing tools of graph grammar to veri...

Let G = (V,E) be a graph. If G is a König graph or if G is a graph without 3-cycles and 5-cycles, we prove that the following conditions are equivalent: ∆G is pure shellable, R/I∆ is Cohen-Macaulay, G is an unmixed vertex decomposable graph and G is well-covered with a perfect matching of König type e1, . . . , eg without 4-cycles with two ei’s. Furthermore, we study vertex decomposable and she...

An orthogonal double cover (ODC) of the complete graph is a collection of graphs such that every two of them share exactly one edge and every edge of the complete graph belongs to exactly two of the graphs. In this paper, we consider the case where the graph to be covered twice is the complete bipartite graph Kmn,mn (for any values ofm,n) and all graphs in the collection are isomorphic to certa...

A graph is called well-covered if every maximal independent set of vertices of G is a maximum independent set; recall that S is independent if no two of its vertices are adjacent. In this paper we define the concept of well-covered by monochromatic paths graphs which is a variation of well-covered graphs. We consider some classical constructions of graphs: G-join of graphs and duplication of a ...

Coverage path planning is the determination of a path that a robot must take in order to pass over each point in an environment. Applications include vacuuming, oor scrubbing, and inspection. We developed the boustrophedon cellular decomposition, which is an exact cellular decomposition approach, for the purposes of coverage. Each cell in the boustrophedon is covered with simple back and forth ...

Let G be a graph of order n. In [10], Saito characterized the graphs with σ3(G) ≥ n− 1 that are not covered by two cycles. In this paper, we characterize the graphs with σ4(G) ≥ n − 1 that are not covered by three cycles. Moreover, to prove our main theorem, we show several new results which are useful in the study of this area.