Quick but Odd Growth of Cacti

Abstract

Let $${\mathcal {F}}$$F be a family of graphs. Given an n-vertex input graph G and a positive integer k, testing whether G has a vertex subset S of size at most k, such that $$G-S$$G-S belongs to $${\mathcal {F}}$$F, is a prototype vertex deletion problem. These type of problems have attracted a lot of attention in recent times in the domain of parameterized complexity. In this paper, we study two such problems; when $${\mathcal {F}}$$F is either the family of forests of cacti or the family of forests of odd-cacti. A graph H is called a forest of cacti if every pair of cycles in H intersect on at most one vertex. Furthermore, a forest of cacti H is called a forest of odd cacti, if every cycle of H is of odd length. Let us denote by $${\mathcal {C}}$$C and $${{\mathcal {C}}}_\mathsf{odd}$$Codd, the families of forests of cacti and forests of odd cacti, respectively. The vertex deletion problems corresponding to $${\mathcal {C}}$$C and $${{\mathcal {C}}}_\mathsf{odd}$$Codd are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with worst case run time $$12^k n^{\mathcal {O}(1)}$$12knO(1) for both these problems. Our algorithms considerably improve the running time for Diamond Hitting Set and Even Cycle Transversal, compared to what is known about them.