The stability number of the graph G, denoted by α(G), is the cardinality of a maximum stable set of G. In this paper we characterize the square-stable graphs, i.e., the graphs enjoying the property α(G) = α(G), where G is the graph with the same vertex set as in G, and an edge of G is joining two distinct vertices, whenever the distance between them in G is at most 2. We show that every square-...

The Shortest Cycle Cover Conjecture asserts that the edges of every bridgeless graph with m edges can be covered by cycles of total length at most 7m/5 = 1.4m. We show that every bridgeless graph with minimum degree three that contains m edges has a cycle cover comprised of three cycles of total length at most 44m/27 ≈ 1.6296m; this extends a bound of Fan [J. Graph Theory 18 (1994), 131–141] fo...

An r-edge-coloring of a graph G is a surjective assignment of r colors to the edges of G. The monochromatic tree partition number of an redge-colored graph G is defined to be the minimum positive integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In this paper, we give a direct proof for the m...

A recent result of Chepoi, Estellon and Vaxes [Disc. Comp. Geom. ’07] states that any planar graph of diameter at most 2R can be covered by a constant number of balls of size R; put another way, there are a constant-sized subset of vertices within which every other vertex is distance half the diameter. We generalize this result to graphs embedded on surfaces of fixed genus with a fixed number o...

A vertex v in a graph G = (V, E) is k-simplicial if the neighborhood N(v) of v can be vertex-covered by k or fewer complete graphs. The main result of the paper states that a planar graph of order at least four has at least four 3simplicial vertices of degree at most five. This result is a strengthening of the classical corollary of Euler’s Formula that a planar graph of order at least four con...

A circuit complexity of a graph is the minimum number of union and intersection operations needed to obtain the whole set of its edges starting from stars. Our main motivation to study this measure of graphs is that it is related to the circuit complexity of boolean functions. We prove some lower bounds to the circuit complexity of explicitly given graphs. In particular, we use the graph theore...

This paper develop a practical and energy-efficient distributed algorithm for the detection of coverage holes in a wireless sensor network. It assumes that the location of sensor nodes is available. We do this in two phases. First we want to identify a set of nodes of which encircle a coverage hole. This is done by introducing a concept of mono-covered arc, which represents the circumference of...

Suppose G = (V,E) is a simple graph and k is a fixed positive integer. A vertex z k-neighborhood-covers an edge (x, y) if d(z, x) ≤ k and d(z, y) ≤ k. A k-neighborhoodcovering set is a set C of vertices such that every edge in E is k-neighborhood-covered by some vertex in C. A k-neighborhood-independent set is a set of edges in which no two distinct edges can be k-neighborhood-covered by the sa...

A 2-edge-covering between G and H is an onto homomorphism from the vertices of G to the vertices of H so that each edge is covered twice and edges in H can be lifted back to edges in G. In this note we show how to compute the spectrum of G by computing the spectrum of two smaller graphs, namely a (modified) form of the covered graph H and another graph which we term the anti-cover. This is done...

A very well–covered graph is an unmixed without isolated vertices such that the height of its edge ideal half number vertices. We study these graphs by means Betti splittings and mapping cone constructions. show cover ideals Cohen–Macaulay are splittable. As a consequence, we compute explicitly minimal graded free resolution class prove have homological linear quotients. Finally, conjecture sam...