Exact Algorithms for Edge Deletion to Cactus

Abstract

We study two related problems on simple, un-directed graphs: Edge Deletion to Cactus and Spanning Tree to Cactus. Edge Deletion to Cactus has been known to be NP-hard on general graphs at least since 1988. We show improved exact algorithms for the former and a polynomial time algorithm for the latter.

Figures (5)

• Figure 1: An edge-minimal non-cactus graph with two marked edges $e_1$ and $e_2$, such that there exists a cycle which contains both $e_1$ and $e_2$, as claimed in \ref{['thm:cycleedges']}
• Figure 2: From left to right: the graph $H$, a spanning cycle $\mathcal{C}$ of $H$, and a spanning cactus subgraph $\mathcal{C}'$ of $H$ with cut vertex $u$, respectively.
• Figure 3: Two distinct cycles $C_1$ and $C_2$ share the path $P$. Removing the edges of $P$ leaves two distinct paths between $u_1$ and $u_3$, which together form a third cycle $C_3$.
• Figure 4: A graph $G$ with the properties as mentioned in the above lemma
• Figure 5: A graph $G$ with the properties as mentioned in the above lemma

Theorems & Definitions (41)

• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• proof
• lemma 1
• proof
• lemma 2
• proof