Distance-critical and distance-redundant graphs

Abstract

If a vertex in a graph can be deleted without affecting distances among the other vertices, we shall say it is distance-redundant. Graphs with all, some or no such vertices are discussed. (The latter class was termed distance-critical by Erdős and Howorka).

Figures (16)

• Figure 1: Illustration of weak link (left) and strong link (right).
• Figure 2: Illustration of redundancy. The left diagram shows a graph $G$. The middle diagram shows $G^*$ which is given by $G$ with edges added between strongly linked vertices. The neighbours of vertex $s$ in $G$ are indicated by filled symbols in $G^*$; the subgraph of $G^*$ induced by these vertices is a complete graph. The right diagram shows $G'$, a graph in which each edge indicates a weak link in $G$. Here the filled vertices induce an empty graph.
• Figure 3: Connected strong graphs of order 5. All can be constructed as described in the proof of lemma \ref{['l.strongsub']}; the first four using weak twins, the last three using strong twins.
• Figure 4: Strong graphs with diameter $n/2$ and no strong twins. There is only one such graph at each even $n$. The first four of an infinite sequence are shown.
• Figure 5: Two graphs with all vertices redundant, none surrounded. The triangular prism is the smallest such graph. The cube is the smallest such graph with diameter above 2.

Theorems & Definitions (24)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.4
• Corollary 2.5
• Theorem 2.6
• Lemma 3.1
• Lemma 3.2
• Corollary 3.3
• Lemma 3.4