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...

We call a vertex x of a graph G = (V,E) a codominated vertex if NG[y] ⊆ NG[x] for some vertex y ∈ V \{x}, and a graph G is called codismantlable if either it is an edgeless graph or it contains a codominated vertex x such that G − x is codismantlable. We show that (C4, C5)-free vertex-decomposable graphs are codismantlable, and prove that if G is a (C4, C5, C7)-free well-covered graph, then ver...

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal inde...

A snake in a graph G is defined to be a closed path in G without proper chords. Let Kd n be the product of d copies of the complete graph Kn. Wojciechowski [13] proved that for any d ≥ 2 the hypercube Kd 2 can be vertex covered with at most 16 disjoint snakes. Alsardary [6] proved that for any odd integer n ≥ 3,d ≥ 2 the graph Kd n can be vertex covered with 2n 3 snakes. We show that for any ev...

We give a formula for the v-number of graded ideal that can be used to compute this number. Then, we show edge I(G) graph G, induced matching number G is an upper bound when very well-covered, or has simplicial partition, well-covered connected and contains neither four, nor five cycles. In all these cases, lower regularity ring G. classify cycle vertices are shedding gain insight into family W...

‎recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎in this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎then we use this construction to generalize the result due to cook and nagel‎.

The notion of removable classes arises in connection with ear decompositions of matching covered graphs introduced by Lovász and Plummer. The last (single or double) ear of an ear decomposition is defined as a removable class. Every matching covered graph not induced by a circuit has at least three removable classes. In this paper, we characterize matching covered graphs with precisely three re...