An animal is an edge connected set of nitely many cells of a regular tiling of the plane. The site-perimeter of an animal is the number of empty cells connected to the animal by an edge. The minimum site-perimeter with a given cell size is found for animals on the triangular and hexagonal grid. The formulas are used to show the e ectiveness of a simple random strategy in full set animal achieve...

We study budget constrained network upgrading problems. We are given an undirected edge weighted graph G V E where node v V can be upgraded at a cost of c v . This upgrade reduces the weight of each edge incident on v. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a minimum spanning tree of weight no more than a given budget D. The results obta...

Let G = (V,E) be a connected graph. G is said to be super edge connected (or super-k for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super-k if for any edge set S # E(G) with jSj 6m, G S is still super-k. The maximum cardinality of m-super-k is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fau...

the degree set of a graph is the set of its degrees. kapoor et al. [degree sets for graphs, fund. math. 95 (1977) 189-194] proved that for every set of positive integers, there exists a graph of diameter at most two and radius one with that degree set. furthermore, the minimum order of such a graph is determined. a graph is 2-self- centered if its radius and diameter are two. in this paper for ...

In this paper we introduce a general strategy for approximating the solution to minimisation problems in random regular graphs. We describe how the approach can be applied to the minimum vertex cover (MVC), minimum independent dominating set (MIDS) and minimum edge dominating set (MEDS) problems. In almost all cases we are able to improve the best known results for these problems. Results for t...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

The k-restricted edge-connectivity of a graph G, denoted by λk(G), is defined as the minimum size of an edge set whose removal leaves exactly two connected components each containing at least k vertices. This graph invariant, which can be seen as a generalization of a minimum edge-cut, has been extensively studied from a combinatorial point of view. However, very little is known about the compl...

A set of vertices S is said to dominate the graph G if for each v / ∈ S, there is a vertex u ∈ S with u adjacent to v. The minimum cardinality of any dominating set is called the domination number of the graph G and is denoted by γ(G). A dominating set D of a graph G = (V,E) is a nonsplit dominating set if the induced graph 〈V − D〉 is connected. The nonsplit domination number γns(G) of the grap...