On the algorithmic complexity of zero-sum edge-coloring

Dehghan, Ali; Sadeghi, Mohammad‐Reza

doi:10.1016/j.ipl.2016.06.010

Cited by 1 publication

(4 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The other parts of the proof are similar to the proof of Theorem 2, Part (ii). It was shown that for a given bipartite (2, 3)-graph G, it is NP-complete to decide whether the graph G has a zero-sum vertex 3-flow [18]. Here, we improve the previous complexity result.…”

Section: Lemmamentioning

confidence: 65%

“…, ±(k − 1)}. It was shown that for a given bipartite (2, 3)-graph G, it is NP-complete to decide whether the graph G has a zero-sum vertex 3-flow [18]. We use from our hardness results on 1-in-Degree decompositions and prove that for a given 3-regular bipartite graph G determining whether G has a zero-sum vertex 3-flow is NP-complete.…”

Section: Our Resultsmentioning

confidence: 99%

“…Problem B. [18] Is there a polynomial time algorithm to decide whether a given graph G has a zero-sum vertex flow?…”

Section: Lemmamentioning

confidence: 99%

“…On the other hand, eigenspaces of the adjacency matrix of graphs have been studied for many years [18,36,40]. This is especially the case for the null-space of the adjacency matrix of graphs, which has been studied for a number of graph classes.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2018

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Lemmamentioning

confidence: 65%

Section: Our Resultsmentioning

confidence: 99%

“…Problem B. [18] Is there a polynomial time algorithm to decide whether a given graph G has a zero-sum vertex flow?…”

Section: Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations