On computing the Hamiltonian index of graphs.

For an integer r ≥ 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 NP -hard for any fixed integer k ≥ 0 even for subcubic graphs G [Ryjáček et al., 2011]. We study the parameterized complexity of this problem with the parameter treewidth, t w (G) , and show that we can find h (G) in time1 O ⋆ ((1 + 2 (ω + 3)) t w (G)) where ω is the matrix multiplication exponent. Prior work on computing h (G) includes various O ⋆ (2 O (t w (G))) -time algorithms for checking if h (G) = 0 holds; i.e., whether G has a Hamiltonian Cycle [Cygan et al., FOCS 2011; Bodlaender et al., Inform. Comput., 2015; Fomin et al., JACM 2016]; an O ⋆ (t w (G) O (t w (G))) -time algorithm for checking if h (G) = 1 holds; i.e., whether L (G) has a Hamiltonian Cycle [Lampis et al., Discrete Appl. Math., 2017]; and, most recently, an O ⋆ ((1 + 2 (ω + 3)) t w (G)) -time algorithm for checking if h (G) = 1 holds [Misra et al., CSR 2019]. Our algorithm for computing h (G) generalizes these results. The 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 Eulerian2 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 O ⋆ ((1 + 2 (ω + 3)) t w (G)) time. To the best of our knowledge this is the first FPT algorithm for Eulerian Steiner Subgraph. Prior work on the special case of finding a spanning Eulerian subgraph (i.e., with K = V (G)) includes a polynomial-time algorithm for series-parallel graphs [Richey et al., 1985] and an O ⋆ (2 O (n)) -time algorithm for planar graphs on n vertices [Sau and Thilikos, 2010]. Our algorithm for Eulerian Steiner Subgraph generalizes both these results. [ABSTRACT FROM AUTHOR]