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Not-all-equal and 1-in-degree decompositions: Algorithmic complexity and applications

Dehghan, A ; Sharif University of Technology | 2018

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  1. Type of Document: Article
  2. DOI: 10.1007/s00453-018-0412-y
  3. Publisher: Springer New York LLC , 2018
  4. Abstract:
  5. 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≥ 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. © 2018, Springer Science+Business Media, LLC, part of Springer Nature
  6. Keywords:
  7. 1-in-Degree decomposition ; Not-All-Equal decomposition ; Total perfect dominating set ; Zero-sum flow ; Zero-sum vertex flow ; Computational complexity ; Matrix algebra ; Parallel processing systems ; Polynomial approximation ; Vector spaces ; Algorithmic complexity ; Bipartite graphs ; Complexity results ; Dominating sets ; Null space ; Polynomial-time algorithms ; Sharp contrast ; Zero sums ; Graph theory
  8. Source: Algorithmica ; Volume 80, Issue 12 , 2018 , Pages 3704-3727 ; 01784617 (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s00453-018-0412-y