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Algorithmic complexity of weakly semiregular partitioning and the representation number

Ahadi, A ; Sharif University of Technology | 2017

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  1. Type of Document: Article
  2. DOI: 10.1016/j.tcs.2017.01.028
  3. Publisher: Elsevier B.V , 2017
  4. Abstract:
  5. 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 degree of T.A graph G is a [d,d+s] -graph if the degree of every vertex of G lies in the interval [d,d+s]. A [d,d+1]-graph is said to be semiregular. The semiregular number of a graph G, denoted by sr(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 semiregular graph. We prove that the semiregular number of a tree T is ⌈δ(T)2⌉. On the other hand, we show that determining whether sr(G)=2 for a given bipartite graph G with δ(G)≤6 is NP-complete.In the second part of the work, we consider the representation number. A graph G has a representation modulo r if there exists an injective map ℓ:V(G)→Zr such that vertices v and u are adjacent if and only if |ℓ(u)-ℓ(v)| is relatively prime to r. The representation number, denoted by rep(G), is the smallest r such that G has a representation modulo r. Narayan and Urick conjectured that the determination of rep(G) for an arbitrary graph G is a difficult problem In this work, we confirm this conjecture and show that if NP≠P, then for any ε(lunate)>0, there is no polynomial time (1-ε(lunate))n2-approximation algorithm for the computation of representation number of regular graphs with n vertices
  6. Keywords:
  7. Edge-partition problems ; Approximation algorithms ; Computational complexity ; Parallel processing systems ; Polynomial approximation ; Set theory ; Trees (mathematics) ; Edge partitions ; Locally irregular graph ; Representation number ; Semiregular number ; Weakly semiregular number ; Graph theory
  8. Source: Theoretical Computer Science ; 2017 ; 03043975 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0304397517301305