3,794 research outputs found
Contracting to a longest path in H-free graphs
The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction
Extremal properties of flood-filling games
The problem of determining the number of "flooding operations" required to
make a given coloured graph monochromatic in the one-player combinatorial game
Flood-It has been studied extensively from an algorithmic point of view, but
basic questions about the maximum number of moves that might be required in the
worst case remain unanswered. We begin a systematic investigation of such
questions, with the goal of determining, for a given graph, the maximum number
of moves that may be required, taken over all possible colourings. We give
several upper and lower bounds on this quantity for arbitrary graphs and show
that all of the bounds are tight for trees; we also investigate how much the
upper bounds can be improved if we restrict our attention to graphs with higher
edge-density.Comment: Final version, accepted to DMTC
Every longest circuit of a 3-connected, -minor free graph has a chord
Carsten Thomassen conjectured that every longest circuit in a 3-connected
graph has a chord. We prove the conjecture for graphs having no
minor, and consequently for planar graphs.Comment: accepted by Journal of Graph Theor
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