In this paper, we continue the study of the domination game in graphs introduced by Brešar, Klavžar, and Rall [SIAM J. Discrete Math. 24 (2010) 979–991]. We study the total version of the domination game and show that these two versions differ significantly. We present a key lemma, known as the Total Continuation Principle, to compare the Dominator-start total domination game and the Staller-st...

All weakly distance-regular digraphs with girth 2 and k − k1,1 = 1 are determined. Moreover, all such digraphs are commutative.

sponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their Hasse digraphs called KoDAGs. KoDAGs are special orderable Directed Acyclic Graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. KoDAGs appear to be distinguished family of Ferrers digraphs which are natural join of a corresponding ordering ch...

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distancetransitive graph. As an application of the result, we obtain some examples of weakly distanceregular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distanc...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...

We resolve two problems of [Cameron, Praeger, and Wormald – Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each ‘building block’ is a finite bipartite graph that is...

The domination number of a graph G = (V,E) is the minimum cardinality of any subset S ⊂ V such that every vertex in V is in S or adjacent to an element of S. Finding the domination numbers of m by n grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomassé. Many variants of domination number on graphs, such as double domination number a...

Given a digraph D and a subset X of vertices of D, pushing X in D means reversing the orientation of all arcs with exactly one end in X. It is known that the problem of deciding whether a given digraph can be made acyclic using the push operation is NP-complete for general digraphs, and polynomial time solvable for multipartite tournaments. Here, we continue the study of deciding whether a digr...

0. Initiated by Sekanina [6], in the sixties and more intensively in the seventies there were considered powers of undirected graphs with special respect to their Hamiltonian behaviour. These investigations have resulted in a lot of interesting and partly very profound propositions; we only remind of the (simple) result of Sekanina [6] that the cube of every finite connected graph is Hamiltonia...

A digraph with maximum out-degree d and radius k has at most 1+d+ · · ·+d vertices, as the Moore bound states. Regular digraphs attaining such a bound and whose diameter is at most k + 1 are called radially Moore digraphs. Knor [4] proved that these extremal digraphs do exist for any value of d ≥ 1 and k ≥ 1. In this paper, we introduce a digraph operator, based on the line digraph, which allow...