On Murty-Simon Conjecture II

TL;DR

This paper proves the Murty-Simon Conjecture for diameter two edge-critical graphs with certain complement properties, confirming the maximum number of edges aligns with the conjectured bound and characterizing extremal graphs.

Contribution

It extends the proof of the Murty-Simon Conjecture to graphs whose complements have specific vertex connectivity and independent vertex cuts.

Findings

Confirmed the conjecture for graphs with complement vertex connectivity 1, 2, 3.

Proved the conjecture for graphs with complements having large independent vertex cuts.

Characterized extremal graphs achieving the maximum edge count.

Abstract

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$. In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al. is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. In this paper, we completely prove the Murty-Simon Conjecture for the graphs whose complements have vertex connectivity $\ell$, where $\ell = 1, 2, 3$; and for the graphs whose complements have an independent vertex cut of cardinality at least three.