Geodetic Number Research Articles

EXTREME STEINER GRAPHS

For a connected graph G of order p ≥ 2 and a set W ⊆ V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W ⊆ V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex (G). A graph G is an extreme Steiner graph if s(G) = ex (G), and an extreme geodesic graph if g(G) = ex (G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 ≤ a ≤ b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l ≥ 2 with r < d ≤ 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 ≤ d < p, 2 ≤ k < p and p - d - k + 2 ≥ 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 ≤ a < b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph.

The 2-edge geodetic number and graph operations

For a connected graph G = (V, E) of order n ≥ 2, a set \({S\subseteq V}\) is a 2-edge geodetic set of G if each edge \({e\in E - E(S)}\) lies on a u-v geodesic with d(u, v) = 2 for some vertices u and v in S. The minimum cardinality of a 2-edge geodetic set in G is the 2-edge geodetic number of G, denoted by eg2(G). It is proved that for any connected graph G, β1(G) ≤ eg2(G), where β1(G) is the matching number of G. It is shown that every pair a, b of integers with 2 ≤ a ≤ b is realizable as the matching number and 2-edge geodetic number, respectively, of some connected graph. We determine bounds for the 2-edge geodetic number of Cartesian product of graphs. Also we determine the 2-edge geodetic number of certain classes of Cartesian product graphs. The 2-edge geodetic number of join of two graphs is obtained in terms of the 2-edge geodetic number of the factor graphs. Open image in new window

The upper connected edge geodetic number of a graph

For a non-trivial connected graph G, a set S ? V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected edge geodetic set of G is the connected edge geodetic number of G and is denoted by g1c(G). A connected edge geodetic set of cardinality g1c(G) is called a g1c- set of G or connected edge geodetic basis of G. A connected edge geodetic set S in a connected graph G is called a minimal connected edge geodetic set if no proper subset of S is a connected edge geodetic set of G. The upper connected edge geodetic number g+ 1c(G) is the maximum cardinality of a minimal connected edge geodetic set of G. Graphs G of order p for which g1c(G) = g+1c = p are characterized. For positive integers r,d and n ( d + 1 with r ? d ? 2r, there exists a connected graph of radius r, diameter d and upper connected edge geodetic number n. It is shown for any positive integers 2 ? a < b ? c, there exists a connected graph G such that g1(G) = a; g1c(G) = b and g+ 1c(G) = c.

Double geodetic number of a graph

Double geodetic number of a graph

On the edge-to-vertex geodetic number of a graph

On the edge-to-vertex geodetic number of a graph

THE LINEAR GEODETIC NUMBER OF A GRAPH

For a connected graph G of order n, an ordered set S = {u1, u2, …, uk} of vertices in G is a linear geodetic set of G if for each vertex x in G, there exists an index i, 1 ≤ i < k such that x lies on a ui - ui + 1 geodesic on G, and a linear geodetic set of minimum cardinality is the linear geodetic number gl(G). The linear geodetic numbers of certain standard graphs are obtained. It is shown that if G is a graph of order n and diameter d, then gl(G) ≤ n - d + 1 and this bound is sharp. For positive integers r, d and k ≥ 2 with r < d ≤ 2r, there exists a connected graph G with rad G = r, diam G = d and gl(G) = k. Also, for integers n, d and k with 2 ≤ d < n, 2 ≤ k ≤ n - d + 1, there exists a connected graph G of order n, diameter d and gl(G) = k. We characterize connected graphs G of order n with gl(G) = n and gl(G) = n - 1. It is shown that for each pair a, b of integers with 3 ≤ a ≤ b, there is a connected graph G with g(G) = a and gl(G) = b. We also discuss how the linear geodetic number of a graph is affected by adding a pendent edge to the graph.

The geodetic number of the lexicographic product of graphs

A set S of vertices of a graph G is a geodetic set if every vertex of G lies in an interval between two vertices from S. The size of a minimum geodetic set in G is the geodetic number g(G) of G. We find that the geodetic number of the lexicographic product G∘H for a non-complete graph H lies between 2 and 3g(G). We characterize the graphs G and H for which g(G∘H)=2, as well as the lexicographic products T∘H that enjoy g(T∘H)=3g(G), when T is isomorphic to a tree. Using a new concept of the so-called geodominating triple of a graph G, a formula that expresses the exact geodetic number of G∘H is established, where G is an arbitrary graph and H a non-complete graph.

On the geodetic and the hull numbers in strong product graphs

A set S of vertices of a connected graph G is convex, if for any pair of vertices u , v ∈ S , every shortest path joining u and v is contained in S . The convex hull C H ( S ) of a set of vertices S is defined as the smallest convex set in G containing S . The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S , and it is said to be a hull set if its convex hull is V ( G ) . The geodetic and the hull numbers of G are the minimum cardinality of a geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also establish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.

The lower and upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u−v path of G, where u, v∈D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori.

The geodetic number of strong product graphs

For two vertices u and v of a connected graph G, the set IG[u; v] consists of all those vertices lying on u v geodesics in G. Given a set S of vertices of G, the union of all sets IG[u; v] for u; v 2 S is denoted by IG[S]. A set S V (G) is a geodetic set if IG[S] = V (G) and the minimum cardinality of a geodetic set is its geodetic number g(G) of G. Bounds for the geodetic number of strong product graphs are obtainted and for several classes improved bounds and exact values are obtained.

The edge geodetic number and Cartesian product of graphs

The edge geodetic number and Cartesian product of graphs

The periphery graph of a median graph

The periphery graph of a median graph is the intersection graph of its peripheral subgraphs. We show that every graph without a universal vertex can be realized as the periphery graph of a median graph. We characterize those median graphs whose periphery graph is the join ∗Work supported by the Ministry of Science of Slovenia and by the Ministry of Science and Technology of India under the bilateral India-Slovenia grants BI-IN/06-07-002 and DST/INT/SLOV-P-03/05, respectively. 18 B. Bresar, M. Changat, A.R. Subhamathi and ... of two graphs and show that they are precisely Cartesian products of median graphs. Path-like median graphs are introduced as the graphs whose periphery graph has independence number 2, and it is proved that there are path-like median graphs with arbitrarily large geodetic number. Peripheral expansion with respect to periphery graph is also considered, and connections with the concept of crossing graph are established.

Some remarks on the geodetic number of a graph

A set of vertices D of a graph G is geodetic if every vertex of G lies on a shortest path between two not necessarily distinct vertices in D . The geodetic number of G is the minimum cardinality of a geodetic set of G . We prove that it is NP-complete to decide for a given chordal or chordal bipartite graph G and a given integer k whether G has a geodetic set of cardinality at most k . Furthermore, we prove an upper bound on the geodetic number of graphs without short cycles and study the geodetic number of cographs, split graphs, and unit interval graphs.

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G . The corresponding remoteness function associates to each vertex x ∈ V ( G ) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G ) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O ( m log n ) time whether G is a median graph with geodetic number 2.

The upper edge geodetic number and the forcing edge geodetic number of a graph

An edge geodetic set of a connected graph \(G\) of order \(p \geq 2\) is a set \(S \subseteq V(G)\) such that every edge of \(G\) is contained in a geodesic joining some pair of vertices in \(S\). The edge geodetic number \(g_1(G)\) of \(G\) is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality \(g_1(G)\) is a minimum edge geodetic set of \(G\) or an edge geodetic basis of \(G\). An edge geodetic set \(S\) in a connected graph \(G\) is a minimal edge geodetic set if no proper subset of \(S\) is an edge geodetic set of \(G\). The upper edge geodetic number \(g_1^+(G)\) of \(G\) is the maximum cardinality of a minimal edge geodetic set of \(G\). The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers \(a\) and \(b\) such that \(2 \leq a \leq b\), there exists a connected graph \(G\) with \(g_1(G)=a\) and \(g_1^+(G)=b\). For an edge geodetic basis \(S\) of \(G\), a subset \(T \subseteq S\) is called a forcing subset for \(S\) if \(S\) is the unique edge geodetic basis containing \(T\). A forcing subset for \(S\) of minimum cardinality is a minimum forcing subset of \(S\). The forcing edge geodetic number of \(S\), denoted by \(f_1(S)\), is the cardinality of a minimum forcing subset of \(S\). The forcing edge geodetic number of \(G\), denoted by \(f_1(G)\), is \(f_1(G) = min\{f_1(S)\}\), where the minimum is taken over all edge geodetic bases \(S\) in \(G\). Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair \(a\), \(b\) of integers with \(0 \leq a \lt b\) and \(b \geq 2\), there exists a connected graph \(G\) such that \(f_1(G)=a\) and \(g_1(G)=b\).

Geodetic sets and Steiner sets in graphs

Suppose that G is a simple graph. Let g ( G ) and s ( G ) be the geodetic number and the Steiner number of G , respectively. In this note, we prove that, for any nonnegative integers a , b and r with a ≥ b ≥ 3 and r ≥ 3 , there exists a connected graph G with g ( G ) = b , s ( G ) = a , and r ( G ) = d ( G ) = r where r ( G ) and d ( G ) are the radius and diameter of G , respectively. This result answers the conjecture of Chartrand and Zhang [G. Chartrand, P. Zhang, The Steiner number of a graph, Discrete Math. 242 (2002) 41–54].

The upper connected geodetic number and forcing connected geodetic number of a graph

For a connected graph G of order p ≥ 2 , a set S ⊆ V ( G ) is a geodetic set of G if each vertex v ∈ V ( G ) lies on an x – y geodesic for some elements x and y in S . The minimum cardinality of a geodetic set of G is defined as the geodetic number of G , denoted by g ( G ) . A geodetic set of cardinality g ( G ) is called a g -set of G . A connected geodetic set of G is a geodetic set S such that the subgraph G [ S ] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g c ( G ) . A connected geodetic set of cardinality g c ( G ) is called a g c -set of G . A connected geodetic set S in a connected graph G is called a minimal connected geodetic set if no proper subset of S is a connected geodetic set of G . The upper connected geodetic number g c + ( G ) is the maximum cardinality of a minimal connected geodetic set of G . We determine bounds for g c + ( G ) and determine the same for some special classes of graphs. For positive integers r , d and n ≥ d + 1 with r ≤ d ≤ 2 r , there exists a connected graph G with rad G = r , diam G = d and g c + ( G ) = n . Also, for any positive integers 2 ≤ a < b ≤ c , there exists a connected graph G such that g ( G ) = a , g c ( G ) = b and g c + ( G ) = c . A subset T of a g c -set S is called a forcing subset for S if S is the unique g c -set containing T . A forcing subset for S of minimum cardinality is a minimum forcing subset of S . The forcing connected geodetic number of S , denoted by f c ( S ) , is the cardinality of a minimum forcing subset of S . The forcing connected geodetic number of G , denoted by f c ( G ) , is f c ( G ) = min { f c ( S ) } , where the minimum is taken over all g c -sets S in G . It is shown that for every pair a , b of integers with 0 ≤ a ≤ b − 4 , there exists a connected graph G such that f c ( G ) = a and g c ( G ) = b .

The hull and geodetic numbers of orientations of graphs

For an oriented graph D , let I D [ u , v ] denote the set of all vertices lying on a u – v geodesic or a v – u geodesic. For S ⊆ V ( D ) , let I D [ S ] denote the union of all I D [ u , v ] for all u , v ∈ S . Let [ S ] D denote the smallest convex set containing S . The geodetic number g ( D ) of an oriented graph D is the minimum cardinality of a set S with I D [ S ] = V ( D ) and the hull number h ( D ) of an oriented graph D is the minimum cardinality of a set S with [ S ] D = V ( D ) . For a connected graph G , let O ( G ) be the set of all orientations of G , define g − ( G ) = min { g ( D ) : D ∈ O ( G ) } , g + ( G ) = max { g ( D ) : D ∈ O ( G ) } , h − ( G ) = min { h ( D ) : D ∈ O ( G ) } , and h + ( G ) = max { h ( D ) : D ∈ O ( G ) } . By the above definitions, h − ( G ) ≤ g − ( G ) and h + ( G ) ≤ g + ( G ) . In the paper, we prove that g − ( G ) < h + ( G ) for a connected graph G of order at least 3, and for any nonnegative integers a and b , there exists a connected graph G such that g − ( G ) − h − ( G ) = a and g + ( G ) − h + ( G ) = b . These results answer a problem of Farrugia in [A. Farrugia, Orientable convexity, geodetic and hull numbers in graphs, Discrete Appl. Math. 148 (2005) 256–262].

The forcing hull and forcing geodetic numbers of graphs

For 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 I G [ u , v ] denote the set of all vertices lying on a u – v geodesic. Let S ⊆ V ( G ) and I G [ S ] denote the union of all I G [ u , v ] for all u , v ∈ S . A subset S ⊆ V ( G ) is a convex set of G if I G [ 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 I G [ 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 f h ( G ) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f g ( G ) of G . In the paper, we construct some 2-connected graph G with ( f h ( G ) , f g ( 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 ( f h ( G ) , f g ( G ) , h ( G ) , g ( G ) ) = ( a , b , a + b + c , a + 2 b + c ) or ( a , 2 a + b , a + b + c , 2 a + 2 b + 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 [formula omitted]-forcing geodetic graphs

For 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 I G [ u , v ] denote the set of all vertices lying on a u - v geodesic, and for S ⊆ V ( G ) , let I G [ S ] denote the union of all I G [ u , v ] for all u , v ∈ S . A set S ⊆ V ( G ) is a geodetic set if I G [ 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 subset of G if there exists a unique minimum geodetic set containing F . A forcing subset F is critical if every proper subset of F is not a forcing subset. The cardinality of a minimum critical forcing subset in G is called the forcing geodetic number f ( G ) of G and the cardinality of a maximum critical forcing subset in G is called the upper forcing geodetic number f + ( G ) of G . If G is a graph with f ( G ) = 0 , then G has a unique minimum geodetic set; that is, f + ( G ) = 0 . In the paper, we prove that, for any nonnegative integers a , b and c with 1 ≤ a ≤ b ≤ c − 2 or 4 ≤ a + 2 ≤ b ≤ c , there exists a connected graph G with f ( G ) = a , f + ( G ) = b , and g ( G ) = c . This result solves a problem of Zhang [P. Zhang, The upper forcing geodetic number of a graph, Ars Combin. 62 (2002) 3–15].