Vertex Covers Revisited: Indirect Certificates and FPT Algorithms

The classical NP-complete problem Vertex Cover requires us to determine whether a graph contains at most $k$ vertices that cover all edges. In spite of its intractability, the problem can be solved in FPT time for parameter $k$ by various techniques. In this paper, we present half a dozen new and simple FPT algorithms for Vertex Cover with respect to parameter $k$. For this purpose, we explore structural properties of vertex covers and use these properties to obtain FPT algorithms by iterative compression, colour coding, and indirect certificating methods. In particular, we show that every graph with a $k$-vertex cover admits an indirect certificate with at most $k\mathrm{/}3$ vertices, which lays the foundation of three new FPT algorithms based on random partition and random selection.