A set D of a vertices in a graph G = (V,E) is said to be a total dominating set of G if every vertex in V is adjacent to some vertex in D. The total domination number γt(G) is the minimum cardinality of a total dominating set. If γt(G) = |V (G)| , the minimum cardinality of a set E0 ⊆ E(G), such that G−E0 contains no isolated vertices and γt(G− E0) > γt(G), is called the total bondage number of...

GP 2 is an experimental nondeterministic programming language based on graph transformation rules, allowing for visual programming and the solving of graph problems at a high-level of abstraction. In previous work we demonstrated how to verify graph programs using a Hoare-style proof calculus, but only partial correctness was considered. In this paper, we add new proof rules and termination fun...

A set D of vertices in a graph G = (V, E) is said to be a total dominating set of G if every vertex in V is adjacent to some vertex in D. The total domination number γt(G) is the minimum cardinality of a total dominating set. E(G) denotes the edge set of G, the complement of G. The minimum cardinality of a set E1 ⊂ E(G) for which γt(G+E1) < γt(G) is denoted by rt(G) and is called the total rein...

Let R be a commutative ring with Z(R) its set of zero-divisors. In this paper, we study the total graph of R, denoted by T(Γ(R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, y ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We study the chromatic number and edge connectivity of this graph.

The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt(G) of a graph G and we show that for any connected graph G of order at least two, msdγt(G) ≤ 3. We show that...

We consider only finite undirected graphs without loops or multiple edges. For terminology and notation not defined here we refer the reader to Bondy and Murty’s textbook [6]. In this note we investigate irregularity measures of graphs. A graph is regular if all its vertices have the same degree, otherwise it is irregular. However, it is of interest to measure how irregular it is. Several appro...

The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme was proved by Bonamy, Charbit and Thomassé. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. Th...

Let H be a hypergraph with vertices V(H) and hyperedges E(H). The total graph of H, T(H), is the simple graph with vertex set V(H)U E(H) where vertices x and y of T(H) are adjacent if and only if x is contained in, contains or is adjacent to y in H. We give a simple characterisation of those graphs which are the total graphs of some hypergraphs. We show that the total graph uniquely defines a l...

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we confirm the total-coloring conjecture for 1-planar graphs with maximum degree