On diameter 2-critical graphs

Minimum vertex-diameter-2-critical graphs

Diameter critical graphs

A graph is called diameter-k-critical if its diameter is k, and the removal of any edge strictly increases the diameter. In this paper, we prove several results related to a conjecture often attributed to Murty and Simon, regarding the maximum number of edges that any diameter-k-critical graph can have. In particular, we disprove a longstanding conjecture of Caccetta and Häggkvist (that in ever...

Regular vertex diameter critical graphs

A graph is called vertex diameter critical if its diameter increases when any vertex is removed. Regular vertex diameter critical graphs of every valency k ≥ 2 and diameter d ≥ 2 exist, raising the question of identifying the smallest such graphs. We describe an infinite family of k-regular vertex diameter critical graphs of diameter d with at most kd+ (2k − 3) vertices. This improves the previ...

On Diameter of Line Graphs

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

Progress on the Murty-Simon Conjecture on diameter-2 critical graphs: a survey

A graph G is diameter 2-critical if its diameter is two and the deletion of any edgeincreases the diameter. Murty and Simon conjectured that the number of edges in adiameter-2-critical graph G of order n is at most bn/4c and that the extremal graphsare the complete bipartite graphs Kbn/2c,dn/2e. We survey the progress made to dateon this conjecture, concentrating mainly on recen...