The additive stretch number of a graph G is the maximum difference of the lengths of a longest induced path and a shortest induced path between two vertices of G that lie in the same component of G.We prove some properties of minimal forbidden configurations for the induced-hereditary classes of graphs G with for some k ∈ N₀ = 0,1,2,.... Furthermore, we derive characterizations of these classes for k = 1 and k = 2.
Dynamic monopolies in graphs have been studied as a model for spreading processes within networks. Together with their dual notion, the generalized degenerate sets, they form the immediate generalization of the classical notions of vertex covers and independent sets in a graph. We present results concerning dynamic monopolies in graphs of given average threshold values extending and generalizing previous results of Khoshkhah et al. [On dynamic monopolies of graphs: The average and strict majority...
Let α ∈ (0,1) and let ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set is called an α-dominating set of G, if for all . We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.
A recent result of Henning and Southey (A note on graphs with disjoint dominating and total dominating set, Ars Comb. 89 (2008), 159-162) implies that every connected graph of minimum degree at least three has a dominating set D and a total dominating set T which are disjoint. We show that the Petersen graph is the only such graph for which D∪T necessarily contains all vertices of the graph.
Let be a set of graphs and for a graph G let and denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on and are presented.
For a graph G, let R(G) and yr2(G) denote the Roman domination number of G and the 2-rainbow domination number of G, respectively. It is known that yr2(G) ≤ R(G) ≤ 3/2yr2(G). Fujita and Furuya [Difference between 2-rainbow domination and Roman domination in graphs, Discrete Appl. Math. 161 (2013) 806-812] present some kind of characterization of the graphs G for which R(G) − yr2(G) = k for some integer k. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs...
We characterize a large subclass of the class of those graphs G for which the exponential domination number of H equals the domination number of H for every induced subgraph H of G.
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