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On the Complexity of Deciding Whether the Regular Number is at Most Two
, Article Graphs and Combinatorics ; Volume 31, Issue 5 , September , 2014 , pp 1359-1365 ; ISSN: 09110119 ; Sadeghi, M. R ; Ahadi, A ; Sharif University of Technology
Abstract
The regular number of a graph (Formula presented.) denoted by (Formula presented.) is the minimum number of subsets into which the edge set of (Formula presented.) can be partitioned so that the subgraph induced by each subset is regular. In this work we answer to the problem posed as an open problem in Ganesan et al. (J Discrete Math Sci Cryptogr 15(2-3):49-157, 2012) about the complexity of determining the regular number of graphs. We show that computation of the regular number for connected bipartite graphs is NP-hard. Furthermore, we show that, determining whether (Formula presented.) for a given connected (Formula presented.)-colorable graph (Formula presented.) is NP-complete. Also, we...
Algorithmic complexity of weakly semiregular partitioning and the representation number
, Article Theoretical Computer Science ; 2017 ; 03043975 (ISSN) ; Dehghan, A ; Mollahajiaghaei, M ; Sharif University of Technology
Elsevier B.V
2017
Abstract
A graph G is weakly semiregular if there are two numbers a,b, such that the degree of every vertex is a or b. The weakly semiregular number of a graph G, denoted by wr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether wr(G)=2 for a given bipartite graph G with at most three numbers in its degree set is NP-complete. Among other results, for every tree T, we show that wr(T)≤2log2 δ(T)+O(1), where δ(T) denotes the maximum...
Algorithmic complexity of weakly semiregular partitioning and the representation number
, Article Theoretical Computer Science ; Volume 674 , 2017 , Pages 60-72 ; 03043975 (ISSN) ; Dehghan, A ; Mollahajiaghaei, M ; Sharif University of Technology
Elsevier B.V
2017
Abstract
A graph G is weakly semiregular if there are two numbers a,b, such that the degree of every vertex is a or b. The weakly semiregular number of a graph G, denoted by wr(G), is the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is a weakly semiregular graph. We present a polynomial time algorithm to determine whether the weakly semiregular number of a given tree is two. On the other hand, we show that determining whether wr(G)=2 for a given bipartite graph G with at most three numbers in its degree set is NP-complete. Among other results, for every tree T, we show that wr(T)≤2log2Δ(T)+O(1), where Δ(T) denotes the maximum...