Steven J. Winters

Directed distance in digraphs: centers and medians

The directed distance dD(u, v) from a vertex u to a vertex v in a strong digraph D is the length of a shortest (directed) u - v path in D. The eccentricity of a vertex v in D is the directed distance from v to a vertex furthest from v. The distance of a vertex v in D is the sum of the directed distances from v to the vertices of D. The center C(D) of D is the subdigraph induced by those vertices of minimum eccentricity, while the median M(D) of D is the subdigraph induced by those vertices of minimum distance. It is shown that for every two asymmetric digraphs D1 and D2, there exists a strong asymmetric digraph H such that C(H) ≅ D1 and M(H) ≅ D2, and where the directed distance from C(H) to M(H) and from M(H) to C(H) can be arbitrarily prescribed. Furthermore, if K is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both D1 and D2, then there exists a strong asymmetric digraph F such that C(F) ≅ D1, M(F) ≅ D2 and C(F) ∩ M(F) ≅ K.

On eccentric vertices in graphs

The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex furthest from v. The minimum eccentricity among the vertices of G is the radius rad G of G, and the maximum eccentricity is its diameter diam G. A vertex u of G is called an eccentric vertex of v if d(u, v) = e(v). The radial number m(v) of v is the minimum eccentricity among the eccentric vertices of v, while the diametrical number dn(v) of v is the maximum eccentricity among the eccentric vertices of v. The radial number m(G) of G is the minimum radial number among the vertices of G and the diametrical number dn(G) of G is the minimum diametrical number among the vertices of G. Several results concerning eccentric vertices are presented. It is shown that for positive integers a and b with a ≤ b ≤ 2a there exists a connected graph G having m(G) = a and dn(G) = b. Also, if a, b, and c are positive integers with a ≤ b ≤ c ≤ 2a, then there exists a connected graph G with rad G = a, m(G) = b, and diam G = c.

γ-labelings of complete bipartite graphs

Explicit formulae for the -min and -max labeling values of complete bipartite graphs are given, along with -labelings which achieve these extremes. A recursive formula for the -min labeling value of any complete multipartite is also presented.

Closed k-stop distance in graphs

The Traveling Salesman Problem (TSP) is still one of the most researched topics in computational mathematics, and we introduce a variant of it, namely the study of the closed k-walks in graphs. We search for a shortest closed route visiting k cities in a non complete graph without weights. This motivates the following definition. Given a set of k distinct vertices S = {x1, x2, . . . , xk} in a simple graph G, the closed k-stop-distance of set S is defined to be dk(S) = min θ∈P(S) ( d(θ(x1), θ(x2))+d(θ(x2), θ(x3))+· · ·+d(θ(xk), θ(x1)) )

Knight’s Tours on Rectangular Chessboards Using External Squares

The classic puzzle of finding a closed knight’s tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight’s tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight’s tour. In this paper we define an extended closed knight’s tour for a rectangular chessboard as a closed knight’s tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight’s tour?

The ultracenter and central fringe of a graph

The central distance of a central vertex v in a connected graph G with rad G < diam G is the largest nonnegative integer n such that whenever x is a vertex with d(v, x) ≤ n then x is also a central vertex. The subgraph induced by those central vertices of maximum central distance is the ultracenter of G. The subgraph induced by the central vertices having central distance 0 is the central fringe of G. For a given graph G, the smallest order of a connected graph H is determined whose ultracenter is isomorphic to G but whose center is not G. For a given graph F, we determine the smallest order of a connected graph H whose central fringe is isomorphic to G but whose center is not G.

Strong Asymmetric Digraphs with Prescribed Interior and Annulus

The directed distance d(u,v) from u to v in a strong digraph D is the length of a shortest u-v path in D. The eccentricity e(v) of a vertex v in D is the directed distance from v to a vertex furthest from v in D. The center and periphery of a strong digraph are two well known subdigraphs induced by those vertices of minimum and maximum eccentricities, respectively. We introduce the interior and annulus of a digraph which are two induced subdigraphs involving the remaining vertices. Several results concerning the interior and annulus of a digraph are presented.

Properties of edge-deleted distance stable graphs

The distance from a vertex u to a vertex v in a connected graph G is the length of a shortest u–v path in G. The distance of a vertex v of G is the sum of the distances from v to the vertices of G. For a vertex v in a 2-edge-connected graph G, we define the edge-deleted distance of v as the maximum distance of v in G −e over all edges e of G. A vertex is an edge-deleted distance stable vertex if the difference between its edge-deleted distance and distance is 1. A 2-edge-connected graph G is an edge-deleted distance stable graph if each vertex of G is an edge-deleted distance stable vertex. In this paper, we investigate the edge-deleted distance of vertices and describe properties of edge-deleted distance stable graphs.

Connected graphs with prescribed median and periphery

Abstract The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex furthest from v in G. The center C(G) of G is the subgraph induced by those vertices of G having minimum eccentricity; the periphery P(G) is the subgraph induced by those vertices of G having maximum eccentricity. The distance d(v) of a vertex v in G is the sum of the distances from v to the vertices of G. The median M(G) of G is the subgraph induced by those vertices having minimum distance. For graphs F and G and a positive integer m, necessary and sufficient conditions are given for F and G to be the median and periphery, respectively, of some connected graph such that the distance between the median and periphery is m. Necessary and sufficient conditions are also given for two graphs to be the median and periphery and to intersect in any common induced subgraph.