Restrained geodetic domination of edge subdivision graph

For a connected graph [Formula: see text], a set [Formula: see text] subset of [Formula: see text] is said to be a geodetic set if all vertices in [Formula: see text] should lie in some [Formula: see text] geodesic for some [Formula: see text]. The minimum cardinality of the geodetic set is the geodetic number. In this paper, the authors discussed the geodetic number, geodetic domination number, and the restrained geodetic domination of the edge subdivision graph.

The edge-to-edge geodetic domination number of a graph

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

On the upper geodetic global domination number of a graph

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

Restrained geodetic domination in graphs

Let [Formula: see text] be a graph with edge set [Formula: see text] and vertex set [Formula: see text]. For a connected graph [Formula: see text], a vertex set [Formula: see text] of [Formula: see text] is said to be a geodetic set if every vertex in [Formula: see text] lies in a shortest path between any pair of vertices in [Formula: see text]. If the geodetic set [Formula: see text] is dominating, then [Formula: see text] is geodetic dominating set. A vertex set [Formula: see text] of [Formula: see text] is said to be a restrained geodetic dominating set if [Formula: see text] is geodetic, dominating and the subgraph induced by [Formula: see text] has no isolated vertex. The minimum cardinality of such set is called restrained geodetic domination (rgd) number. In this paper, rgd number of certain classes of graphs and 2-self-centered graphs was discussed. The restrained geodetic domination is discussed in graph operations such as Cartesian product and join of graphs. Restrained geodetic domination in corona product between a general connected graph and some classes of graphs is also discussed in this paper.

Geodetic global domination in corona and strong product of graphs

A set S of vertices in a connected graph [Formula: see text] is called a geodetic set if every vertex not in [Formula: see text] lies on a shortest path between two vertices from [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] has at least one neighbor in [Formula: see text]. A set [Formula: see text] is called a geodetic global dominating set of [Formula: see text] if [Formula: see text] is both geodetic and global dominating set of [Formula: see text]. The geodetic global domination number is the minimum cardinality of a geodetic global dominating set in [Formula: see text]. In this paper, we determine the geodetic global domination number of the corona and strong products of two graphs.

The forcing near geodetic number of a graph

A set [Formula: see text] is a near geodetic set if for every [Formula: see text] in [Formula: see text] there exist some [Formula: see text] in [Formula: see text] with [Formula: see text] The near geodetic number [Formula: see text] is the minimum cardinality of a near geodetic set in [Formula: see text] A subset [Formula: see text] of a minimum near geodetic set [Formula: see text] is called the forcing subset of [Formula: see text] if [Formula: see text] is the unique minimum near geodetic set containing [Formula: see text]. The forcing number [Formula: see text] of [Formula: see text] in [Formula: see text] is the minimum cardinality of a forcing subset for [Formula: see text], while the forcing near geodetic number [Formula: see text] of [Formula: see text] is the smallest forcing number among all minimum near geodetic sets of [Formula: see text]. In this paper, we initiate the study of forcing near geodetic number of connected graphs. We characterize graphs with [Formula: see text]. Further, we compare the parameters geodetic number[Formula: see text] near geodetic number[Formula: see text] forcing near geodetic number and we proved that, for every positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a nontrivial connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text].

Geodetic Global Domination in the Join of Two Graphs

A set S of vertices in a connected graph is called a geodetic set if every vertex not in lies on a shortest path between two vertices from . A set of vertices in is called a dominating set of if every vertex not in has at least one neighbor in . A set is called a geodetic global dominating set of if is both geodetic and global dominating set of . The geodetic global dominating number is the minimum cardinality of a geodetic global dominating set in . In this paper we determine the geodetic global domination number of the join of two graphs.

Further results on the outer connected geodetic number of a graph

For a connected graph G of order at least two, an outer connected geodetic set S in a connected graph G is called a minimal outer connected geodetic set if no proper subset of S is an outer connected geodetic set of G. The upper outer connected geodetic number g+ oc(G) of G is the maximum cardinality of a minimal outer connected geodetic set of G. We determine bounds for it and find the upper outer connected geodetic number of some standard graphs. Some realization results on the upper outer connected geodetic number of a graph are studied. The proposed method can be extended to the identification of beacon vertices towards the network fault-tolerant in wireless local access network communication. Also, another parameter forcing outer connected geodetic number fog(G) of a graph G is introduced and several interesting results and realization theorem are proved.

Convexidade em Grafo Linha de Bipartido

For a nontrivial connected and simple graphs G= (V(G), E(G)), a set S E(G) is called edge geodetic set of G if every edge of G it’s in S or is contained in a geodesic joining some pair of edges in S. The edge geodetic number eds(G) of G is the minimum order of its edge geodetic sets. We prove that it is NP-complete to decide for a given bipartiti graphs G and a given integer k whether G has a edge geodetic set of cardinality at most k. A set M V(G) is called P3 set of G if all vertices of G have two neighbors in M. The P3 number of G is the minimum order of its P3 sets. We prove that it is NP-complete to decide for a given graphs G(diamond,odd-hole) free and a given integer k whether G has a P3 set of cardinality at most k.