# Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications

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## Abstract

A Not-All-Equal 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, $$r\ge 3$$, for a given r-regular bipartite graph G determining whether G has a 1-in-Degree decomposition is $$\mathbf {NP}$$-complete. These complexity results have been especially useful in proving $$\mathbf {NP}$$-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 $$\{ \pm \, 1,\pm \,2\}$$ is $$\mathbf {NP}$$-complete. Among other results, we introduce a new version of Planar 1-in-3 SAT and we prove that this version is also $$\mathbf {NP}$$-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 $$\{ \pm \,1,\pm \,2\}$$ is $$\mathbf {NP}$$-complete.

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## Acknowledgements

The authors would like to thank the anonymous referees for their useful comments which helped to improve the presentation of this paper.

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Dehghan, A., Sadeghi, MR. & Ahadi, A. Not-All-Equal and 1-in-Degree Decompositions: Algorithmic Complexity and Applications. Algorithmica 80, 3704–3727 (2018). https://doi.org/10.1007/s00453-018-0412-y

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• DOI: https://doi.org/10.1007/s00453-018-0412-y