Exterior Vertices in Graphs and Realization of Plurality Preference Digraphs.

Abstract

Part I: Peripheral and eccentric vertices of graphs. For a connected graph G, let P(G) and EC(G) denote the sets of peripheral vertices and eccentric vertices of G, respectively. In 1988 F. Buckley initiated the study of the class S of graphs G for which P(G) = EC(G). We provide several families of graphs which are in S. For certain graphs G with diameter equal to 2r(G) or 2r(G) or 2r(G) $-$ 1, we give criteria for P(G) and EC(G) to be equal. Also, for certain products, we characterize those pairs of graphs so that their product is in S. We present graphs which are then used to show that all possible set-inclusion relations between P(G) and EC(G) may occur. Additionally, we estimate the smallest order of a graph H having a given graph G as an induced subgraph so that P(H) = EC(H) (or P(H) = V(G)). Part II: Realization of plurality preference digraphs. A digraph D with vertex set $\{x\sb1,x\sb2,\...,x\sb{k}\}$ is (n,h,k)-realizable if there exists a connected graph G of order n, a subset V $\subseteq$ V(G) with $\vert V\vert$ = h, and a subset C = $\{c\sb1,c\sb2,\...,c\sb{k}\}\subseteq V(G)$ so that for all distinct i and j in $\{1,2,\...,k\}$, ($x\sb{i},x\sb{j})$ is an arc of D if and only if more vertices in V are closer to $c\sb{j}$ in G. In particular, if h = n, then we simply say that D is realizable by G or that G realizes D. In 1988, Johnson and Slater proved that any oriented graph is realizable by a graph. We provide two constructions of graphs which realize a given oriented graph and show that each of these has a smaller order than the example due to Johnson and Slater. The best known construction, due recently to by W. Schnyder is also provided. Secondly, we characterize digraphs which are realizable by trees. Additionally, we derive some properties of a digraph which is (n,n,n)-realizable by a tree and describe a class of such digraphs. Finally, let ${\cal F}\sb{n}$ denote the family of digraphs of order n which are realizable by trees. For a fixed D $\in$ ${\cal F}\sb{n}$, let $\alpha$(D) be the smallest order of a tree which realizes D. We determine the value of $\alpha$(${\cal F}\sb{n}$ = $max\{\alpha(D)$: $D \in\ {\cal F}\sb{n}\}$ explicitly.