let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

let $g=(v,e)$ be a simple graph. a set $dsubseteq v$ is adominating set of $g$ if every vertex in $vsetminus d$ has atleast one neighbor in $d$. the distance $d_g(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$g$. an $(u,v)$-path of length $d_g(u,v)$ is called an$(u,v)$-geodesic. a set $xsubseteq v$ is convex in $g$ ifvertices from all $(a, b)$-geodesics belon...

Let $G=(V,E)$ be a simple graph. A set $Dsubseteq V$ is adominating set of $G$ if every vertex in $Vsetminus D$ has atleast one neighbor in $D$. The distance $d_G(u,v)$ between twovertices $u$ and $v$ is the length of a shortest $(u,v)$-path in$G$. An $(u,v)$-path of length $d_G(u,v)$ is called an$(u,v)$-geodesic. A set $Xsubseteq V$ is convex in $G$ ifvertices from all $(a, b)$-geodesics belon...

For a connected graph G = (V,E), a set D ⊆ V (G) is a dominating set of G if every vertex in V (G)−D has at least one neighbour in D. The distance dG(u, v) between two vertices u and v is the length of a shortest (u− v) path in G. An (u− v) path of length dG(u, v) is called an (u− v)-geodesic. A set X ⊆ V (G) is convex in G if vertices from all (a − b)-geodesics belong to X for any two vertices...

A set D of vertices in a graph G is a dominating set if every vertex of G, which is not in D, has a neighbor in D. A set of vertices D in G is convex (respectively, isometric), if all vertices in all shortest paths (respectively, all vertices in one of the shortest paths) between any two vertices in D lie in D. The problem of finding a minimum convex dominating (respectively, isometric dominati...

In a graph G = (V, E), a set D ⊂ V is a weak convex dominating(WCD) set if each vertex of V-D is adjacent to at least one vertex in D and d < D > (u, v) = d G (u, v) for any two vertices u, v in D. A weak convex dominating set D, whose induced graph < D > has no cycle is called acyclic weak convex dominating(AWCD) set. The domination number γ ac (G) is the smallest order of a acyclic weak conve...

A bipartite graph G = (X,Y ;E) is convex if there exists a linear enumeration L of the vertices of X such that the neighbours of each vertex of Y are consecutive in L. We show that the problems of finding a minimum dominating set and a minimum independent dominating set in an n-vertex convex bipartite graph are solvable in time O(n2). This improves previous O(n3) algorithms for these problems. ...

A number of optimization methods require as a rst step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an in nite number of) convex functions, and we show how to o...

A subset of vertices of a graph is called a dominating set if every vertex of the graph which is not present in the set has at least one neighbour in it. Dominating set polytope of a graph is defined as the convex hull of 0/1-incidence vectors of all the dominating sets of the graph. This paper presents complete characterization of the dominating set polytope of a cycle.

for allx ∈X and all T ∈ . Since the appearance of Grothendieck-Pietsch’s domination theorem for p-summing operators, there is a great interest in finding out the structure of uniformly dominated sets. We will denote by p(μ) the set of all operators T ∈ Πp(X,Y) satisfying (1.1) for all x ∈ X. It is easy to prove that p(μ) is absolutely convex, closed, and bounded (for the p-summing norm). In [4]...