Quick but Odd Growth of Cacti

Abstract

Let F be a family of graphs. Given an input graph G and a positive integer k, testing whether G has a k-sized subset of vertices S, such that G \ S belongs to 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 F is either a family of cactus graphs or a family of odd-cactus graphs. A graph H is called a cactus graph if every pair of cycles in H intersect on at most one vertex. Furthermore, a cactus graph H is called an odd cactus, if every cycle of H is of odd length. Let us denote by C and C odd , families of cactus and odd cactus, respectively. The vertex deletion problems corresponding to C and C odd are called Diamond Hitting Set and Even Cycle Transversal, respectively. In this paper we design randomized algorithms with running time 12 k n O(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.

Feedback Vertex Set and Even Cycle Transversal for $H$-Free Graphs: Finding Large Block Graphs

“…We aim to extend known complexity results for Feedback Vertex Set for H-free graphs and to perform a new, similar study for Even Cycle Transversal (for which, so far, mainly parameterized complexity results exist [2,3,11,12]). To describe the known and new results we need some terminology.…”

An Improved Deterministic Parameterized Algorithm for Cactus Vertex Deletion

“…For instance, Pseudo Forest Vertex Deletion can be solved deterministically in time O * (3 k ) [2] and randomizedly in time O * (2.85 k ) [12] and Bounded Degree-2 Vertex Deletion can be solved deterministically in time O * (3.0645 k ) [19] and randomizedly in time O * (3 k ) [8]. Among others, the known gap on Cactus Vertex Deletion is remarkable: Bonnet et al [3] presented a deterministic O * (26 k )-time algorithm, while Koley et al [15] presented a randomized O * (12 k )-time algorithm.…”

“…Note that a graph has no cycles of even length if and only if it is a forest of odd cacti [15]. Kolet et al [15] gave an O * (12 k )-time randomized algorithm and Misra et al [17] gave an O * (50 k )-time deterministic algorithm for Even Cycle Transversal. In this paper, we improve the running time of the deterministic algorithm for Even Cycle Transversal.…”

Parameterized Vertex Deletion Problems for Hereditary Graph Classes with a Block Property

“…For instance, a forest is a graph in the class Φ {K2} , a cactus graph is a graph in the class Φ C where C consists of K 2 and all cycles, and a complete-block imply that there is a c k n O(1) -time algorithm for Diamond Hitting Set [9,12,14], but an exact value for c is not forthcoming from these approaches. However, Kolay et al [15] obtained a 12 k n O(1) -time randomized algorithm. For the variant where each cycle must additionally be odd (that is, P consists of K 2 and all odd cycles), there is a 50 k n O (1) -time deterministic algorithm due to Misra et al [18].…”