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The forwarding indices of graphs - a survey

Xu J.-M., Xu M.

Opuscula Mathematica

2013

Vol. 33, no. 2

345--372

A routing R of a connected graph G of order n is a collection of n(n — 1) simple paths connecting every ordered pair of vertices of G. The vertex-forwarding index £(G,R) of G with respect to a routing R is defined as the maximum number of paths in R passing through any vertex of G. The vertex-forwarding index £(G) of G is defined as the minimum £(G,R) over all routings R of G. Similarly, the edge-forwarding index π (G,R) of G with respect to a routing R is the maximum number of paths in R passing through any edge of G. The edge-forwarding index π (G) of G is the minimum π(G, R) over all routings R of G. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some difficulty and relations to other topics in graph theory.

A note on the p-domination number of trees

Lu Y., Hou X., Xu J.-M.

Opuscula Mathematica

2009

Vol. 29, no. 2

157-164

Let p be a positive integer and G = (V (G), E(G)) a graph. A p-dominating set of G is a subset S of V (G) such that every vertex not in S is dominated by at least p vertices in S. The p-domination number ϒp(G) is the minimum cardinality among the p-dominating sets of G. Let T be a tree with order n ≥ 2 and p ≥ 2 a positive integer. A vertex of V (T) is a p-leaf if it has degree at most p - 1, while a p-support vertex is a vertex of degree at least p adjacent to a p-leaf. In this note, we show that ϒp(T) ≥ (n + /Lp(T)/ - /Sp(T)/)/2, where Lp(T) and Sp(T) are the sets of p-leaves and p-support vertices of T, respectively. Moreover, we characterize all trees attaining this lower bound.