On Computing the Hamiltonian Index of Graphs

Abstract

For an integer $r \ge 0$ the $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest $r$ such that $L^{r}(G)$ has a Hamiltonian cycle [Chartrand, 1968]. Checking if $h(G) = k$ is $\mathsf {NP}$-hard for any fixed integer $k \ge 0$ even for subcubic graphs $G$ [Ryjáček et al., 2011]. We study the parameterized complexity of this problem with the parameter treewidth, $tw(G)$, and show that we can find $h(G)$ in time $\mathcal {O}^{\star }((1 + 2^{(\omega + 3)})^{tw(G)})$ where $\omega$ is the matrix multiplication exponent. This generalizes various prior results on computing $h(G)$ including an $\mathcal {O}^{\star }((1 + 2^{(\omega + 3)})^{tw(G)})$-time algorithm for checking if $h(G) = 1$ holds [Misra et al., CSR 2019].

The $\mathsf {NP}$-hard Eulerian Steiner Subgraph problem takes as input a graph $G$ and a specified subset $K$ of terminal vertices of $G$ and asks if $G$ has an Eulerian subgraph $H$ containing all the terminals. A key ingredient of our algorithm for finding $h(G)$ is an algorithm which solves Eulerian Steiner Subgraph in $\mathcal {O}^{\star }((1 + 2^{(\omega + 3)})^{tw(G)})$ time. To the best of our knowledge this is the first $\mathsf {FPT}$ algorithm for Eulerian Steiner Subgraph, and generalizes previous results on various special cases.

Full version on arXiv: https://arxiv.org/abs/1912.01990.