On Computing the Hamiltonian Index of Graphs ⋆

The $r$-th iterated line graph $L^{r}(G)$ of a $G$ 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 $G$. Hamiltonian Index $h(G)$ smallest $r$ such that has cycle. Checking if $h(G) k$ NP-hard any fixed integer $k \geq 0$ even subcubic graphs We study parameterized complexity this problem with parameter treewidth, $tw(G)$, show we can find in time $O*((1 + 2^{(\omega 3)})^{tw(G)})$ $\omega$ matrix multiplication exponent $O*$ notation hides polynomial factors input size. The Eulerian Steiner Subgraph takes as specified subset $K$ terminal vertices asks an (that is: connected, all degree.) subgraph $H$ containing terminals. A second result (and key ingredient our algorithm finding $h(G)$) work which solves time.