A note on graphs with diameter‐preserving spanning trees

Abstract

The distance between a pair of vertices u, u in a graph G is the length of a shortest path joining u and u. The diameter diam(G) of G is the maximum distance between all pairs of vertices in G. A spanning tree Tof G is diameter preserving if diam(T) = diam(G). In this note, we characterize graphs that have diameter-preserving spanning trees.The distance d,(u, u ) between a pair of vertices u and u in a graph G is the length of a shortest path joining u and u. If u and u are in different components of G , then d,(u, u) = x . The eccentricity ec(u) of vertex u in G is its distance to a farthest vertex. When the graph in question is clear, we simply write d (u, u ) and e(u), omitting the subscript. If u is a vertex such that d(u, u ) = e(u), then u is called an eccentric vertex of u . The diameter, diam(G), and the radius, r(G), are the maximum and minimum eccentricities, respectively. The center C ( G ) is the set of vertices with minimum eccentricity and the periphery is the set of vertices with maximum eccentricity. Thus, a center vertex has e(u) = r ( G ) , while a peripheral vertex has e(u) = diam(G).Lesniak [2] characterized sequences of integers that are realizable as the sequence of eccentricities of a graph. More recently, Nandakumar [3] characterized graphs G with a spanning tree T that preserves the eccentricities, i.e., eT(u) = ec(u) for each vertex u E G as follows:

Monophonic Distance in Graphs

“…For any two vertices u and v in G, the distance d (u, v) from u to v is defined as the length of a shortest u − v path in G. The eccentricity e(v) of a vertex v in G is the maximum distance from v to a vertex of G. The radius rad G of G is the minimum eccentricity among the vertices of G, while the diameter diam G of G is the maximum eccentricity among the vertices of G. The distance between two vertices is a fundamental concept in pure graph theory, and this distance is a metric on the vertex set of G. More results related to this distance are found in Refs. [1][2][3][4][5][6][7][8][9]. This distance is used to study the central concepts like center, median, and centroid of a graph [10][11][12][13][14][15][16][17][18][19][20][21][22].…”

Spanning trees with disjoint 2 dominating and 2-dominating sets

“…Observe that not every DD 2 -graph has a spanning DD 2 -tree -as an example can serve the subdivision graph of any corona graph having a cycle (see formal definitions below). Therefore, our problem perfectly fits into the bunch of problems related to the concept of spanning trees having the same or approximately the same properties as the input graph, i.e., center-preserving spanning trees [3], diameter-preserving spanning trees [4,5,15], degree-preserving spanning trees [2,14], and t-spanners [6,17].…”

Adaptive Network Layer

“…The papers [83][84][85][86][87][88][89][90][91][92][93][94][95][96] are concerned with some aspects of graph connectivity other than the usual path oriented one, primarily with those deriving from the notion of the diameter of a graph (i.e., the maximum node to node hop distance across the graph) or the average node to node hop distance. Of special interest here is the notion of leverage, as described in the papers of Bagga et al [83], which is a general method of quantifying changes in graph invariants due to the loss of some network components.…”