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.

Additional Metadata
Keywords Edge-partition problems, Locally irregular graph, Representation number, Semiregular number, Weakly semiregular number
Persistent URL dx.doi.org/10.1016/j.tcs.2017.01.028
Journal Theoretical Computer Science
Citation
Ahadi, A. (Arash), Dehghan, A. (Ali), & Mollahajiaghaei, M. (Mohsen). (2017). Algorithmic complexity of weakly semiregular partitioning and the representation number. Theoretical Computer Science, 674, 60–72. doi:10.1016/j.tcs.2017.01.028