The total Steiner number of a graph

Abstract

<jats:p> A total Steiner set of [Formula: see text] is a Steiner set [Formula: see text] such that the subgraph [Formula: see text] induced by [Formula: see text] has no isolated vertex. The minimum cardinality of a total Steiner set of [Formula: see text] is the total Steiner number of [Formula: see text] and is denoted by [Formula: see text]. Some general properties satisfied by this concept are studied. Connected graphs of order [Formula: see text] with total Steiner number 2 or 3 are characterized. We partially characterized classes of graphs of order [Formula: see text] with total Steiner number equal to [Formula: see text] or [Formula: see text] or [Formula: see text]. It is shown that [Formula: see text]. It is shown that for every pair k, p of integers with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text]. Also, it is shown that for every positive integer [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text], there exists a connected graph [Formula: see text] of order [Formula: see text] such that [Formula: see text] and [Formula: see text]. </jats:p>

The total edge Steiner number of a graph

“…The minimum cardinality of a total Steiner set of G is the total Steiner number of G and is denoted by s t (G). The total Steiner number of a graph was introduced in [10]. Steiner trees have application in transportation networks.…”

“…Theorem 1.1. [10,16] Each extreme vertex of a graph G belongs to every edge Steiner set (total Steiner set) of G. In particular, each end-vertex of G belongs to every edge Steiner set (total Steiner set) of G.…”

“…Theorem 1.2. [10] Let G is a connected graph and v an extreme vertex of G. Then every total Steiner set of G contains at least one element of N (v). Remark 2.2.…”