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.

Highlights

  • In the field of parameterized graph algorithms, vertex deletion problems constitute a considerable fraction

  • 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

  • If F is a family of edgeless graphs, or forests or bipartite graphs, it corresponds to Vertex Cover, Feedback Vertex Set, and Odd Cycle Transversal, respectively

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Summary

Introduction

In the field of parameterized graph algorithms, vertex (edge) deletion (addition, editing) problems constitute a considerable fraction. Misra et al [15] used the structural properties of an odd-cactus graph to design an algorithm for Even Cycle Transversal with running time 50knO(1) They give an O(k2) kernel for the problem. To apply our reduction rules in a way that this fraction is as small as possible, we study a more general problem than Even Cycle Transversal, which we call Parity Even Cycle Transversal In this problem we are given a graph G and a weight function w : E(G) → {0, 1} and the objective is to delete a subset S of vertices of size at most k such. We conclude the introduction by noting that Diamond Hitting Set and Even Cycle Transversal admit approximation algorithms with factor 9 and 10 respectively [6, 15]

Preliminaries
Counting Lemma
Algorithm for Even Cycle Transversal
Algorithm for Diamond Hitting Set
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