On Computing the Hamiltonian Index of Graphs
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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. Checking if h(G) = k is NP-hard for any fixed integer k ≥ 0 even for subcubic graphs G. We study the parameterized complexity of this problem with the parameter treewidth, tw(G), and show that we can find h(G) in time O*((1 + 2^(ω + 3))^tw(G)) where ω is the matrix multiplication exponent and the O* notation hides polynomial factors in input size. 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 Eulerian (that is: connected, and with all vertices of even degree.) subgraph H containing all the terminals. A second result (and a key ingredient of our algorithm for finding h(G)) in this work is an algorithm which solves Eulerian Steiner Subgraph in O*((1 + 2^(ω + 3))^tw(G)) time.
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