Outer-convex domination in graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] of vertices of a graph [Formula: see text] is an outer-convex dominating set if every vertex not in [Formula: see text] is adjacent to some vertex in [Formula: see text] and [Formula: see text] is a convex set. The outer-convex domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-convex dominating set of [Formula: see text]. An outer-convex dominating set of cardinality [Formula: see text] will be called a [Formula: see text]-[Formula: see text]. In this paper, we initiate the study and characterize the outer-convex dominating sets in the join of the two graphs.

Weakly convex domination subdivision number of a graph

A set X is weakly convex in G if for any two vertices a,b ? X there exists an ab-geodesic such that all of its vertices belong to X. A set X ? V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number ?wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd?wcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.

Convex dominating-geodetic partitions in graphs

The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. A u-v path of length d(u,v) is called u-v geodesic. A set X is convex in G if vertices from all a -b geodesics belong to X for every two vertices a,b?X. A set of vertices D is dominating in G if every vertex of V-D has at least one neighbor in D. The convex domination number con(G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a x-y geodesic between two vertices x,y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by g(G). Let D,S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a ?con(G)-set and S is a g(G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.