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We consider computational complexity questions related to parallel knock-out schemes for graphs. In such schemes, in each
round, each remaining vertex of a given graph eliminates exactly one of its neighbours. We show that the problem of whether,
for a given graph, such a scheme can be found that eliminates every vertex is NP-complete. Moreover, we...
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Citations
A parallel knock-out scheme for a graph proceeds in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is KO-reducible if there exists such a scheme that eliminates every vertex in the graph. The Parallel Knock-Out problem is to decide whether a graph G is KO-reducible. This problem is known to be NP-complete and has been studied for several graph classes since MFCS 2004. We show that the problem is NP-complete even for split graphs, a subclass of P
5-free graphs. In contrast, our main result is that it is linear-time solvable for P
4-free graphs (cographs).
We study parallel knock-out schemes for graphs. These schemes proceed in rounds in each of which each surviving vertex simultaneously
eliminates one of its surviving neighbours; a graph is reducible if such a scheme can eliminate every vertex in the graph.
We show that, for a reducible graphG, the minimum number of required rounds is
O(Ö{a})O{({\sqrt{\alpha}})}
, where α is the independence number of G. This upper bound is tight and the result implies the square-root conjecture which was first posed in MFCS 2004. We also
show that for reducible K
1,p
-free graphs at most p − 1 rounds are required. It is already known that the problem of whether a given graph is reducible is NP-complete. For claw-free graphs, however, we show that this problem can be solved in polynomial time.
In 1997 Lampert and Slater introduced parallel knock-out schemes, an iterative process on graphs that goes through several rounds. In each round of this process, every vertex eliminates exactly one of its neighbors. The parallel knock-out number of a graph is the minimum number of rounds after which all vertices have been eliminated (if possible). The parallel knock-out number is related to well-known concepts like perfect matchings, hamiltonian cycles, and 2-factors.We derive a number of combinatorial and algorithmic results on parallel knock-out numbers: for families of sparse graphs (like planar graphs or graphs of bounded tree-width), the parallel knock-out number grows at most logarithmically with the number n of vertices; this bound is basically tight for trees. Furthermore, there is a family of bipartite graphs for which the parallel knock-out number grows proportionally to the square root of n. We characterize trees with parallel knock-out number at most 2, and we show that the parallel knock-out number for trees can be computed in polynomial time via a dynamic programming approach (whereas in general graphs this problem is known to be NP-hard). Finally, we prove that the parallel knock-out number of a claw-free graph is either infinite or less than or equal to 2.
An H1,{H2}H1,{H2}-factor of a graph GG is a spanning subgraph of GG with exactly one component isomorphic to the graph H1H1 and all other components (if there are any) isomorphic to the graph H2H2. We completely characterise the class of connected almost claw-free graphs that have a P7,{P2}P7,{P2}-factor, where P7P7 and P2P2 denote the paths on seven and two vertices, respectively. We apply this result to parallel knock-out schemes for almost claw-free graphs. These schemes proceed in rounds in each of which each surviving vertex eliminates one of its surviving neighbours. A graph is reducible if such a scheme eliminates every vertex in the graph. Using our characterisation, we are able to classify all reducible almost claw-free graphs, and we can show that every reducible almost claw-free graph is reducible in at most two rounds. This leads to a quadratic time algorithm for determining if an almost claw-free graph is reducible (which is a generalisation and improvement upon the previous strongest result that showed that there was a O(n5.376)O(n5.376) time algorithm for claw-free graphs on nn vertices).
We study parallel knock-out schemes for graphs. These schemes proceed in rounds in each of which each surviving vertex simultaneously eliminates one of its surviving neighbours; a graph is reducible if such a scheme can eliminate every vertex in the graph. We resolve the square-root conjecture, first posed at MFCS 2004, by showing that for a reducible graph G, the minimum number of required rounds is O(n); in fact, our result is stronger than the conjecture as we show that the minimum number of required rounds is O(α), where α is the independence number of G. This upper bound is tight. We also show that for reducible K 1,p -free graphs at most p-1 rounds are required. It is already known that the problem of whether a given graph is reducible is NP-complete. For claw-free graphs, however, we show that this problem can be solved in polynomial time. We also pinpoint a relationship with (locally bijective) graph homomorphisms.
