Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications
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A Not-All-Equal (NAE) decomposition of a graph G is a decomposition of the vertices of G into two parts such that each vertex in G has at least one neighbor in each part. Also, a 1-in-Degree decomposition of a graph G is a decomposition of the vertices of G into two parts A and B such that each vertex in the graph G has exactly one neighbor in part A. Among our results, we show that for a given graph G, if G does not have any cycle of length congruent to 2 mod 4, then there is a polynomial time algorithm to decide whether G has a 1-in-Degree decomposition. In sharp contrast, we prove that for every r, r≥ 3, for a given r-regular bipartite graph G determining whether G has a 1-in-Degree decomposition is NP -complete. These complexity results have been especially useful in proving NP -completeness of various graph related problems for restricted classes of graphs. In consequence of these results we show that for a given bipartite 3-regular graph G determining whether there is a vector in the null-space of the 0,1-adjacency matrix of G such that its entries belong to {± 1,± 2} is NP -complete. Among other results, we introduce a new version of Planar 1-in-3 SAT and we prove that this version is also NP -complete. In consequence of this result, we show that for a given planar (3,4)-semiregular graph G determining whether there is a vector in the null-space of the 0,1-incidence matrix of G such that its entries belong to {± 1,± 2} is NP -complete.
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