Edge, vertex and mixed fault diameters

Abstract

Let ▫${mathcal{D}}^E_q(G)$▫ denote the maximum diameter among all subgraphs obtained by deleting ▫$q$▫ edges of ▫$G$▫. Let ▫${mathcal{D}}^V_p(G)$▫ denote the maximum diameter among all subgraphs obtained by deleting ▫$p$▫ vertices of ▫$G$▫. We prove that ▫${mathcal{D}}^E_a(G) leqslant {mathcal{D}}^V_a(G) + 1$▫ a for all meaningful ▫$a$▫. We also define mixed fault diameter ▫${mathcal{D}}^M_{(p,q)}(G)$▫, where ▫$p$▫ vertices and ▫$q$▫ edges are deleted at the same time. We prove that for ▫$0 < l leqslant a$▫, ▫${mathcal{D}}^E_a(G) leqslant {mathcal{D}}^M_{(a-ell,ell)}(G) leqslant {mathcal{D}}^V_a(G) + 1$▫, and give some examples

Bounds on the fault‐diameter of graphs

“…For triangle-free graphs we improve these bounds to 3n−1 5 for k = 2 and to approximately 2 k+1 n for k ≥ 3. Our bounds on the k-edge-fault-diameter for k ≥ 3 strengthen the above-mentioned bounds (1) and (2) in the sense that they show that these bounds on the diameter of graphs with minimum degree δ still hold asymptotically after removal of up to δ − 1 edges, provided the graph is δ-edge-connected.…”

“…Hence there are at least 2 (1,2) incidences between edges of S and vertices of V 1 (V 2 , V 3 ), which adds up to a total of 5 incidences of vertices in V and edges in S. But there are exactly 4 incidences of the two edges of S with vertices of V, a contradiction. Hence we have at most one 1-pair.…”

“…Han and Zhang determined the maximum size of a graph of given order and k ‐fault‐diameter. Banic, Erves, and Zerovnik showed that in a ‐connected graph the k ‐fault‐diameter can exceed the k ‐edge‐fault‐diameter by at most one. For results on the (edge‐)fault‐diameter of Cartesian products of graphs and their generalisations see, for example, .…”

Mixed fault diameter of Cartesian graph bundles II

“…In previous work [6] on vertex, edge and mixed fault diameters of connected graphs the following theorem has been proved. Theorem 2.7.…”

Wide-diameter of Product Graphs

“…If some nodes or links are faulty, information may not be transmitted by these nodes and links. Therefore graphs with low fault diameters with respect to vertex faults [1,18], edge faults [2], and mixed faults [3] yield interesting interconnection networks. In [19], an upper bound for (vertex) fault-diameter of graph product is given in terms of fault diameters of the base and the fibre.…”