Non-empty Set Of Vertices Research Articles

The Steiner diameter of a graph with prescribed girth

Let G be a connected graph of order p, and let S be a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2≤n≤p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. We give upper bounds on the Steiner n-diameter of G in terms of order, minimum degree δ, and girth g, of G. Moreover, we construct graphs to show that, if for given δ and g there exists a Moore graph of minimum degree δ and girth g, then the bounds are asymptotically sharp.

Decomposable Convexities in Graphs and Hypergraphs

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

On vertices enforcing a Hamiltonian cycle

A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.

Formula omitted]-force sets of locally semicomplete digraphs

Let D be a hamiltonian digraph. A nonempty vertex set X⊆V(D) is called an H-force set of D if every X-cycle of D (i.e. a cycle of D containing all vertices of X) is hamiltonian. The H-force numberh(D) of a digraph D is defined to be the smallest cardinality of an H-force set of D. In this paper, the minimal H-force sets of locally semicomplete digraphs are characterized and the H-force number is given.

Upper bounds on the Steiner diameter of a graph

Let G be a connected graph of order p and S a nonempty set of vertices of G. Then the Steiner distance d(S) of S is the minimum size of a connected subgraph of G whose vertex set contains S. If n is an integer, 2≤n≤p, the Steiner n-diameter, diamn(G), of G is the maximum Steiner distance of any n-subset of vertices of G. We give a bound on diamn(G) for a graph G in terms of the order of G and the minimum degree of G. Our result implies a bound on the ordinary diameter by Erdős, Pach, Pollack and Tuza. We obtain improved bounds on diamn(G) for K3-free graphs and C4-free graphs. Moreover, we construct graphs to show that the bounds are asymptotically best possible.

Small alliances in a weighted graph

We extend the notion of a defensive alliance to weighted graphs. Let ( G , w ) be a weighted graph, where G is a graph and w be a function from V ( G ) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑ x ∈ N G ( v ) ∩ S w ( x ) + w ( v ) ≥ ∑ x ∈ N G ( v ) − S w ( x ) holds for every vertex v in S . Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most ⌈ 1 2 n ⌉ . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n . Then we prove that for any weight function w on V ( G ) , ( G , w ) has a defensive weighted alliance of order at most ⌈ 1 2 n ⌉ . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3].

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G . A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S . Let k ≥ 2 be an integer. A set X of vertices of G is k -Steiner convex if it contains the Steiner interval of every set of k vertices in X . A vertex x ∈ X is an extreme vertex of X if X ∖ { x } is also k -Steiner convex. We call such vertices k -Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.

Computing the Omega polynomial of an infinite family of the linear parallelogram P(n,m)

Let G=(V,E) be a molecular graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges. The Omega polynomial Ω(G,x) was introduced by Diudea in 2006 and this defined as where m(G,c) the number of qoc strips of length c. In this paper, we compute the omega polynomial of an infinite family of the linear parallelogram P(n,n) of benzenoid graph.

Fibonacci Polynomials and Parity Domination in Grid Graphs

A non-empty set of vertices is called an even dominating set if each vertex in the graph is adjacent to an even number of vertices in the set (adjacency is reflexive). In this paper, the Fibonacci polynomials are studied over GF(2) with particular emphasis on their divisibility properties and their relation to the existence of even dominating sets in grid graphs and properties of a corresponding recurrence.

On the Steiner median of a tree

Let S be a nonempty set of vertices of a connected graph G. Then the Steiner distance of S is the minimum size (number of edges) of a connected subgraph of G containing S. Let n ⩾ 2 be an integer and suppose that G has at least n vertices. Then the Steiner n-distance of a vertex v of G is defined to be the sum of the Steiner distances of all sets of n vertices that include v. The Steiner n-median M n ( G) of G is the subgraph induced by the vertices of minimum Steiner n-distance. We show that the Steiner n-median of a tree is connected and determine those trees that are Steiner n-medians of trees, and show that if T is a tree with more than n vertices, then M n ( T) ⊂- M n + 1 ( T). Further, a O(¦V(T)¦) algorithm for finding the Steiner n-median of a tree T is presented and a O(n¦V(T)¦ 2) algorithm for finding the Steiner n-distances of all vertices in tree T is described. For a connected graph G of order p ⩾ n, the n-median value of G is the least Steiner n-distance of any of its vertices. For p ⩾ 2 n − 1, sharp upper and lower bounds for the n-median values of trees of order p are given, and it is shown that among all trees of a given order, the path has maximum n-median value.

Steiner distance stable graphs

Let G be a connected graph and S a nonempty set of vertices of G. Then the Steiner distance d G ( S) of S is the smallest number of edges in a connected subgraph of G that contains S. Let k, l, s and m be nonnegative integers with m⩾ s⩾2 and k and l not both 0. Then a connected graph G is said to be k-vertex l-edge ( s, m)-Steiner distance stable, if for every set S of s vertices of G with d G ( S)= m, and every set A consisting of at most k vertices of G- S and at most l edges of G, d G− A ( S)= d G ( S). It is shown that if G is k-vertex l-edges ( s, m)-Steiner distance stable , then G is k-vertex l-edge ( s, m+1)-Steiner distance stable. Further, a k-vertex l-edge ( s, m)-Steiner distance stable graph is shown to be a k-vertex l-edge ( s−1, m)-Steiner distance stable graph for s⩾3. It is then shown that the converse of neither of these two results holds. If G is a connected graph and S an independent set of s vertices of G such that d G ( S)= m, then S is called an I( s, m)-set. A connected graph is k-vertex l-edge I( s, m)-Steiner distance stable if for every I( s, m)-set S and every set A of at most k vertices of G- S and l-edges of G, d G− A ( S)= m. It is shown that a k-vertex l-edge I(3, m)-Steiner distance stable graph, m⩾4, is k-vertex l-edge I(3, m+1)-Steiner distance stable.

On finite fixed sets in infinite graphs

An automorphism of a graph X is called a translation of X if it fixes no finite non-empty set of vertices of X. It is shown that a group G of automorphisms of the connected graph X fixes a finite non-empty set of vertices or ends of X if and only if any two translations of X in G have a common fixed end. Applications and refinements are discussed.

Extended Chromatic Polynomials

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.