On computing the Hamiltonian index of graphs

Abstract

For an integer r≥0 the r-th iterated line graphLr(G) of a graph G is defined by: (i) L0(G)=G and (ii) Lr(G)=L(L(r−1)(G)) for r>0, where L(G) denotes the line graph of G. The Hamiltonian Indexh(G) of G is the smallest r such that Lr(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, tw(G), and show that we can find h(G) in time1O⋆((1+2(ω+3))tw(G)) where ω is the matrix multiplication exponent. Prior work on computing h(G) includes various O⋆(2O(tw(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⋆(tw(G)O(tw(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))tw(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))tw(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⋆(2O(n))-time algorithm for planar graphs on n vertices [Sau and Thilikos, 2010]. Our algorithm for Eulerian Steiner Subgraph generalizes both these results.