A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} such that every vertex v ∈ V with f(v) = 0 has at least one neighbor u ∈ V with f(u) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γR(G). The Roman bondage number...

I-language over simple type structure is considered. Type Y== (0 --t (0 -+ 0)) + (0 -+ 0) is called a binary tree type because of the isomorphism between binary trees and closed terms of this type. Therefore any closed term of type Y+(Y-P ... -P (Y + Y) ) represents an n-ary tree function. The problem is to characterize tree operations represented by the closed terms of the examined type. It is...

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g−1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less ...

A graph whose edges are labeled either as positive or negative is called a signed graph. A signed graph is said to be net-regular if every vertex has constant net-degree k, namely, the difference between the number of positive and negative edges incident with a vertex. In this paper, we analyze some properties of co-regular signed graphs which are net-regular signed graphs with the underlying g...

The base of a signed digraph S is the minimum number k such that for any vertices u, v of S, there is a pair of walks of length k from u to v with different signs. Let K be a signed complete graph of order n, which is a signed digraph obtained by assigning +1 or −1 to each arc of the n-th order complete graph Kn considered as a digraph. In this paper we show that for n ≥ 3 the base of a primiti...