The forcing hull and forcing geodetic numbers of graphs

AbstractFor every pair of vertices u,v in a graph, a u–v geodesic is a shortest path from u to v. For a graph G, let IG[u,v] denote the set of all vertices lying on a u–v geodesic. Let S⊆V(G) and IG[S] denote the union of all IG[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if IG[S]=S. A convex hull [S]G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if IG[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number fh(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number fg(G) of G. In the paper, we construct some 2-connected graph G with (fh(G),fg(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (fh(G),fg(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81–94]

The Forcing Geodetic Cototal Domination Number of a Graph

Let  be a geodetic cototal domination set of . A subset  is called a forcing subset for  if  is the unique minimum geodetic cototal domination set containing . The minimum cardinality T is the forcing geodetic cototal domination number of S is denotedby , is the cardinality of a minimum forcing subset of S. The forcing geodetic cototal domination number of ,denoted by , is , where the minimum is takenover all -sets  in . Some general properties satisfied by this concept arestudied. It is shown that for every pair  of integers with ,there exists a connected graph  such that  and . where  isthe geodetic cototal dominating number of

Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

On the Forcing Hull and Forcing Monophonic Hull Numbers of Graphs

For a connected graph G = (V,E), let a set M be a minimum monophonic hull set of G. A subset T ⊆ M is called a forcing subset for M if M is the unique minimum monophonic hull set containing T

The restrained monophonic number of a graph

A set S of vertices of a connected graph G is a monophonic set of G if each vertex v of G lies on a x−y monophonic path for some x and y in S. The minimum cardinality of a monophonic set of G is the monophonic number of G and is denoted by m(G). A restrained monophonic set S of a graph G is a monophonic set such that either S = V or the subgraph induced by V − S has no isolated vertices. The minimum cardinality of a restrained monophonic set of G is the restrained monophonic number of G and is denoted by mr(G). We determine bounds for it and determine the same for some special classes of graphs. Further, several interesting results and realization theorems are proved.Publisher's Versio

THE RESTRAINED STEINER NUMBER OF A GRAPH

For a connected graph G = (V, E) of order p, a set W ⊆ V is called a Steiner set of G if every vertex of G is contained in a Steiner W-tree of G. The Steiner number s(G) of G is the minimum cardinality of its Steiner sets. A set W of vertices of a graph G is a restrained Steiner set if W is a Steiner set, and if either W = V or the subgraph G[V − W ] induced by V − W has no isolated vertices. The minimum cardinality of a restrained Steiner set of G is the restrained Steiner number of G, and is denoted by s r (G). The restrained Steiner number of certain classes of graphs are determined. Connected graphs of order p with restrained Steiner number 2 are characterized. Various necessary conditions for the restrained Steiner number of a graph to be p are given. It is shown that, for integers a, b and p with 4 ≤ a ≤ b ≤ p, there exists a connected graph G of order p such that s(G) = a and s r (G) = b. It is also proved that for every pair of integers a, b with a ≥ 3 and b ≥ 3, there exists a connected graph G with s r (G) = a and g r (G) = b

Algorithms to Find Linear Geodetic Numbers and Linear Edge Geodetic Numbers in Graphs

Given two vertices u and v of a connected graph G=(V, E), the closed interval I[u, v] is that set of all vertices lying in some u-v geodesic in G. A subset of V(G) S={v1,v2,v3,….,vk} is a linear geodetic set or sequential geodetic set if each vertex x of G lies on a vi – vi+1 geodesic where 1 ? i < k . A linear geodetic set of minimum cardinality in G is called as linear geodetic number lgn(G) or sequential geodetic number sgn(G). Similarly, an ordered set S={v1,v2,v3,….,vk} is a linear edge geodetic set if for each edge e = xy in G, there exists an index i, 1 ? i < k such that e lies on a vi – vi+1 geodesic in G. The cardinality of the minimum linear edge geodetic set is the linear edge geodetic number of G denoted by legn(G). The purpose of this paper is to introduce algorithms using dynamic programming concept to find minimum linear geodetic set and thereby linear geodetic number and linear edge geodetic set and number in connected graphs

Symbolic regression for approximating graph geodetic number

Graph properties are certain attributes that could make the structure of the graph understandable. Occasionally, standard methods cannot work properly for calculating exact values of graph properties due to their huge computational complexity, especially for real-world graphs. In contrast, heuristics and metaheuristics are alternatives proved their ability to provide sufficient solutions in a reasonable time. Although in some cases, even heuristics are not efficient enough, where they need some not easily obtainable global information of the graph. The problem thus should be dealt in completely different way by trying to find features that related to the property and based on these data build a formula which can approximate the graph property. In this work, symbolic regression with an evolutionary algorithm called Cartesian Genetic Programming has been used to derive formulas capable to approximate the graph geodetic number which measures the minimal-cardinality set of vertices, such that all shortest paths between its elements cover every vertex of the graph. Finding the exact value of the geodetic number is known to be NP-hard for general graphs. The obtained formulas are tested on random and real-world graphs. It is demonstrated how various graph properties as training data can lead to diverse formulas with different accuracy. It is also investigated which training data are really related to each property