The Deck Ratio and Self-Repairing Graphs

Abstract

In this paper, we define, for a graph invariant \(\psi\), the deck ratio of \(\psi\) by \(D_\psi(G) = \frac{\psi(G)}{\Sigma_{v\in V(G)}\psi(G-v)}\). We give generic upper and lower bounds on \(D_\psi\) for monotone increasing and monotone decreasing invariants \(\psi\), respectively.

Then, we proceed to consider the Wiener index \(W(G)\), showing that \(D_W(G) \leq \frac{1}{|V(G)|-2}\). We show that equality is attained for a graph \(G\) if and only if every induced \(P_3\) subgraph of \(G\) is contained in a \(C_4\) subgraph. Such graphs have been previously studied under the name of self-repairing graphs.

We show that a graph on \(n \geq 4\) vertices with at least \(\frac{n^2-3n+6}{2} – n + 3\) edges is necessarily a self-repairing graph and that this is the best possible result. We also show that a \(2\)-connected graph is self-repairing if and only if all factors in its Cartesian product decomposition are.

Finally, some open problems about the deck ratio and about self-repairing graphs are posed at the end of the paper.