The /-connectivity ~c/(G) of a graph G of order p ~ > / I is the minimum number of vertices that are required to be deleted from G to produce a graph with at least f components or with fewer than /' vertices. Extremal graphs are investigated and bounds established on the number of edges in graphs of given order and f-connectivity. @ 1999 Elsevier Science B.V, All rights reserved

A new graph parameter, edge neighbor toughness is introduced to measure how difficult it for a be broken into many components through the deletion of closed neighborhoods few edges. Let G=(V,E) graph. An e said subverted when its neighborhood and two endvertices are deleted from G. set S⊆E(G) called an cut-strategy if all edges in S has been G survival subgraph, denoted by G/S, disconnected, or...

We prove upper and lower bounds on the size of the largest square grid graph that is a subgraph, minor, or shallow minor of a graph in the form of a larger square grid from which a specified number of vertices have been deleted. Our bounds are tight to within constant factors. We also provide less-tight bounds on analogous problems for higher-dimensional grids.

The connected components of an undirected graph are determined in the EREW model with p processors, using a parallel extension of the “sparsification” technique. A previous algorithm due to Kruskal et al. is optimal forp < ml/(*+&) < n/2. We show that our algorithm is work-optimal for an extended range of values. Using the “sparsification” tree data structure we provide also a dynamic algorithm...

Given a graph G = (V, E), with each subset X of V is associated the subgraph G(X) of G induced by X. A subset I of V is an interval of G provided that for any a, b ∈ I and x ∈ V \ I , {a, x} ∈ E if and only if {b, x} ∈ E. For example, ∅, {x}, where x ∈ V , and V are intervals of G called trivial intervals. A graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable...

A -dimensional -square matrix is defined and certain properties of such matrices are investigated. Two particular graph constructions involving -dimensional -square matrices are given and the graphs so constructed are called matrix graphs. Properties of matrix graphs are determined and an application of matrix graphs to completely independent critical clique is provided. Some attention is g...

A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e ∈ E(G). In this paper we show that for any integers k ≥ 3 and l ≥ 5 there exists a constant c = c(k, l) > 0, such that for all ñ, there exists a (k + 1)-critical graph G on n vertices with n > ñ and odd girth at least l, which can be made (k − 1)-colourable only by the omission of at least cn2 edges.

We improve the best known bounds on the average degree of online k-list-critical graphs for k > 6. Specifically, for k > 7 we show that every non-complete online k-list-critical graph has average degree at least k−1+ (k−3) 2(2k−3) k4−2k3−11k2+28k−14 and every non-complete online 6-list-critical graph has average degree at least 5 + 93 766 . The same bounds hold for offline k-list-critical graphs.