A Characterization of Connected (1,2)-Domination Graphs of Tournaments

Recently. Hedetniemi et aI. introduced (1,2)-domination in graphs, and the authors extended that concept to (1, 2)-domination graphs of digraphs. Given vertices x and y in a digraph D, x and y form a (1,2)-dominating pair if and only if for every other vertex z in D, z is one step away from x or y and at most two steps away from the other. The (1,2)-dominating graph of D, dom1,2 (D), is defined to be the graph G = (V, E ) , where V (G) = V (D), and xy is an edge of G whenever x and y form a (1,2)-dominating pair in D. In this paper, we characterize all connected graphs that can be (I, 2)-dominating graphs of tournaments

Sign-nonsingular matrices and even cycles in directed graphs

AbstractA sign-nonsingular matrix or L-matrix A is a real m× n matrix such that the columns of any real m×n matrix with the same sign pattern as A are linearly independent. The problem of recognizing square L-matrices is equivalent to that of finding an even cycle in a directed graph. In this paper we use graph theoretic methods to investigate L-matrices. In particular, we determine the maximum number of nonzero elements in square L-matrices, and we characterize completely the semicomplete L-matrices [i.e. the square L-matrices (aij) such that at least one of aij and aij is nonzero for any i,j] and those square L-matrices which are combinatorially symmetric, i.e., the main diagonal has only nonzero entries and aij=0 iff aji=0. We also show that for any n×n L-matrix there is an i such that the total number of nonzero entries in the ith row and ith column is less than n unless the matrix has a completely specified structure. Finally, we discuss the algorithmic aspects

Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating 2-broadcasts along with the associated parameter, the dominating 2-broadcast number. We prove that computing the dominating 2-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.Peer ReviewedPostprint (author's final draft