On distances in Sierpiński graphs: Almost-extreme vertices and metric dimension

Abstract

Sierpi?ski graphs Sn p form an extensively studied family of graphs of fractal nature applicable in topology, mathematics of the Tower of Hanoi, computer science, and elsewhere. An almost-extreme vertex of Sn p is introduced as a vertex that is either adjacent to an extreme vertex of Sn p or is incident to an edge between two subgraphs of Sn p isomorphic to Snp-1. Explicit formulas are given for the distance in Sn p between an arbitrary vertex and an almostextreme vertex. The formulas are applied to compute the total distance of almost-extreme vertices and to obtain the metric dimension of Sierpi?ski graphs.

On the roman domination number of generalized Sierpiński graphs

“…The idea of almost-extreme vertex of S(K n , t) was introduced in [15] as a vertex that is either adjacent to an extreme vertex of S(K n , t) or is incident to an edge between two subgraphs of S(K n , t) isomorphic to S(K n , t−1). The authors of [15] deduced explicit formulas for the distance in S(K n , t) between an arbitrary vertex and an almost-extreme vertex. Also they gave a formula of the metric dimension of a Sierpiński graph, which was independently obtained by Parreau in her Ph.D. thesis.…”

The strong metric dimension of generalized Sierpiński graphs with pendant vertices

“…Later, those graphs have been called Sierpiński graphs in [7] and they were studied by now from numerous points of view. The reader is invited to read, for instance, the following recent papers [2,5,4,7,8,9] and references therein. This construction was generalized in [3] for any graph G, by defining the t-th generalized Sierpiński graph of G, denoted by S(G, t), as the graph with vertex set V t (G) and edge set defined as follows.…”

Generalized Power Domination: Propagation Radius and Sierpiński Graphs

“…See recent books [14,24] for many connections between Sierpiński graphs and these two topics and [15] for the problem of when Sierpiński graphs embed as spanning subgraphs into the corresponding Tower of Hanoi graphs. In addition, Sierpiński graphs were studied from numerous other points of view, recent investigations include [11,16,17,20,23,28,29,30]. We also point out that earlier than the Sierpiński graphs, the so-called WK-recursive networks were introduced in [6], see also [10].…”