Let $R$ be a commutative ring and $M$ be an $R$-module with $T(M)$ as subset, the set of torsion elements. The total graph of the module denoted by $T(Gamma(M))$, is the (undirected) graph with all elements of $M$ as vertices, and for distinct elements $n,m in M$, the vertices $n$ and $m$ are adjacent if and only if $n+m in T(M)$. In this paper we study the domination number of $T(Gamma(M))$ a...

Two new domination parameters for a connected graph G: the weakly convex domination number of G and the convex domination number of G are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number.

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

We study convexity properties of graphs. In this paper we present a linear-time algorithm for the geodetic number in tree-cographs. Settling a 10-year-old conjecture, we prove that the Steiner number is at least the geodetic number in AT-free graphs. Computing a maximal and proper monophonic set in AT-free graphs is NP-complete. We present polynomial algorithms for the monophonic number in perm...

A set of vertices S in a graph is called geodetic if every vertex of this graph lies on some shortest path between two vertices from S. In this paper, minimum geodetic sets in median graphs are studied with respect to the operation of peripheral expansion. Along the way geodetic sets of median prisms are considered and median graphs that possess a geodetic set of size two are characterized. © 2...

Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes vk, vl ∈ S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S∗ such that I[S∗] = V . In “The geodetic number of a graph”, Mathematical and Computer Modelling 17 (June 1993) 8...

A set S of vertices of a connected graph G is convex, if for any pair of vertices u, v ∈ S , every shortest path joining u and v is contained in S . The convex hull CH(S ) of a set of vertices S is defined as the smallest convex set in G containing S . The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex ...

In this paper, we survey some new results in four areas of domination in graphs, namely: (1) the toughness and matching structure of graphs having domination number 3 and which are “critical” in the sense that if one adds any missing edge, the domination number falls to 2; (2) the matching structure of graphs having domination number 3 and which are “critical” in the sense that if one deletes a...