## Dissertations

#### Title

Integrity of Digraphs

12-1996

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Lowell Beineke

Dr. Gary Chartrand

Dr. Yousef Alavi

Dr. John Petro

#### Abstract

The vertex-integrity of a digraph D, denoted I(D), is defined to be the minimum over aIII subsets X of the vertex set of D for the quantity IXI + m(D - X), w here IXI is the number of vertices in X and m(D - X) is the maximum order of a strong component in the digraph D - X. In a like manner, the arc-integrity of the digraph D, denoted I’(D), is defined to be the minimum over all subsets Y of the arc set of D for the quantity IYI + m(D - Y), where IYI is the number of arcs in Y. These two measures of the vulnerability of a digraph are analogous to the undirected concepts, which w ere introduced by Barefoot, Entringer and Swart in 1987.

This investigation of these two parameters centers on the vertex-integrity and arc-integrity for orientations of graphs in several interesting families, including complete graphs, complete bipartite graphs, cartesian products of paths, and hypercubes. Because different orientations of the same graph may lead to different values of the parameters, we can only hope to bound these values for a given graph or class of graphs. Every graph has an acyclic orientation, where the largest strong component is of order 1 . Both the vertex-integrity and arc-integrity for such an orientation are 1. This being the case, we focus on the maximum vertex-(arc-)integrity which can be attained by some orientation of the graph.

If one can find an induced subgraph H of a graph G for which any orientation of H has vertex-integrity 1, then the maximum vertex-integrity which an orientation could attain is at most IG - H I +- 1. To that end, we define the decycling number of a graph G, denoted V(G), to be the minimum order of a subset S of the vertices of G, such that G - S is a forest Since G - S is acyclic, then for any orientation of G - S, the order of the largest strong component is 1. Therefore the maxim um vertex-integrity over all orientations of G is at most (G) + 1. This parameter is investigated for the families of graphs we study in the chapter on vertex-integrity.