Revisiting Path Contraction and Cycle Contraction

Abstract

The Path Contraction and Cycle Contraction problems take as input an undirected graph G with n vertices, m edges and an integer k and determine whether one can obtain a path or a cycle, respectively, by performing at most k edge contractions in G. We revisit these NP-complete problems and prove the following results.

• Path Contraction admits an $\mathcal {O}^*(2^{k})$-time algorithm. This improves over the current algorithm known for the problem [Algorithmica 2014].

• Cycle Contraction admits an $\mathcal {O}^*((2 + \epsilon _{\ell })^k)$-time algorithm where $0 < \epsilon _{\ell } \le 0.5509$ and $\epsilon _{\ell }$ decreases as length of the resultant cycle $\ell$ increases.

Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal {O}^*(2.5191^n)$-time algorithm to solve the optimization version of Cycle Contraction.

Next, we turn our attention to restricted graph classes and show the following results.

• Path Contraction on planar graphs admits a polynomial-time algorithm.

• Path Contraction on chordal graphs does not admit an $\mathcal {O}(n^{2-\epsilon } \cdot 2^{o(tw)})$-time algorithm for any $\epsilon > 0$, unless the Orthogonal Vectors Conjecture fails. Here, tw is the treewidth of the input graph.

The second result complements the $\mathcal {O}(nm)$-time, i.e., $\mathcal {O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014].

R. Krithika—Supported by SERB MATRICS grant number MTR/2022/000306.