Domination Critical Knödel Graphs

Abstract

A set D of vertices of a graph G is a dominating set if each vertex of V (G) \ D, is adjacent to some vertex of D. The domination number of G, γ(G), is the minimum cardinality of a dominating set of G. A graph G is called domination vertex critical, or just γ-critical if removal of any vertex decreases the domination number. A graph G is called domination vertex stable, or just γ-stable, if removal of any vertex does not decrease the domination number. For an even integer n ≥ 2 and 1 ≤ ∆ ≤ ⌊log 2 n⌋, a Knödel graph W ∆,n is a ∆-regular bipartite graph of even order n, with vertices (i, j), for i = 1, 2 and 0 ≤ j ≤ n/2 − 1, where for every j, 0 ≤ j ≤ n/2 − 1, there is an edge between vertex (1, j) and every vertex (2, (j + 2 k − 1) mod (n/2)), for k = 0, 1, · · · , ∆−1. In this paper, we study the domination criticality and domination stability of Knödel graphs. We characterize the 3-regular and 4-regular Knödel graphs by γcriticality or γ-stability.

2-power domination number for Knödel graphs and its application in communication networks

“…Mojdeh et al discussed the value of total domination for cubic Knödel graph in [23]. Domination number for 4-regular Knödel graph [21], and domination number for critical Knödel graph [22] have also been determined. Varghese et al discussed the power domination number for Knödel graphs in [29], and the details along with the results obtained in this paper are listed in Table 1.…”

Domination in 4-regular Knödel graphs

“…In this section we review some properties in the Knödel graphs that are proved in [7]. Mojdeh et al considered a re-labeling on the vertices of a Knödel graph as follows: we label (1, i) by u i+1 for each i = 0, 1, ..., n/2 − 1, and (2, j) by v j+1 for j = 0, 1, ..., n/2 − 1.…”

“…They obtained exact domination number for W 3,n . Mojdeh et al [7] determined the total domination number in 3-regular Knödel graphs W 3,n . In this paper, we determine the domination number in 4-regular Knödel graphs W 4,n .…”

Disjunctive total domination stability in graphs

“…Then, Brigham et al 5 initialized the study of the graphs where the domination number decreases on the removal of any vertex. Those graphs are called as vertex domination critical 6–8 . After then, total domination critical graphs were defined by Goddard et al 9 and studied in some papers 10–12 .…”

“…Those graphs are called as vertex domination critical. [6][7][8] After then, total domination critical graphs were defined by Goddard et al 9 and studied in some papers. [10][11][12] Likewise, the critical concept is applied to other variations of domination such as independent domination, 12 -domination, 13 distance domination, 14 and paired domination.…”