Almost all k-cop-win graphs contain a dominating set of cardinality k

TL;DR

This paper proves that almost all k-cop-win graphs in a random graph contain a dominating set of size k, extending known results about cop-win graphs with a universal vertex.

Contribution

It generalizes the known property of cop-win graphs to k-cop-win graphs, showing they almost all contain a dominating set of size k.

Findings

Almost all k-cop-win graphs contain a dominating set of size k.

Asymptotic enumeration of labelled k-cop-win graphs provided.

Extension of known properties from cop-win graphs to k-cop-win graphs.

Abstract

We consider $k$-cop-win graphs in the binomial random graph $G(n,1/2).$ It is known that almost all cop-win graphs contain a universal vertex. We generalize this result and prove that for every $k \in N$, almost all $k$-cop-win graphs contain a dominating set of cardinality $k$. From this it follows that the asymptotic number of labelled $k$-cop-win graphs of order $n$ is equal to $(1+o(1)) (1-2^{-k})^{-k} {n \choose k} 2^{n^2/2 - (1/2-\log_2(1-2^{-k})) n}$.