Decomposing Cubic Graphs into Connected Subgraphs of Size Three

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some cases.Comment: to appear in the proceedings of COCOON 201

Edge-partitioning regular graphs for ring traffic grooming with a priori placement od the ADMs

We study the following graph partitioning problem: Given two positive integers C and Δ, find the least integer M(C,Δ) such that the edges of any graph with maximum degree at most Δ can be partitioned into subgraphs with at most C edges and each vertex appears in at most M(C,Δ) subgraphs. This problem is naturally motivated by traffic grooming, which is a major issue in optical networks. Namely, we introduce a new pseudodynamic model of traffic grooming in unidirectional rings, in which the aim is to design a network able to support any request graph with a given bounded degree. We show that optimizing the equipment cost under this model is essentially equivalent to determining the parameter M(C, Δ). We establish the value of M(C, Δ) for almost all values of C and Δ, leaving open only the case where Δ ≥ 5 is odd, Δ (mod 2C) is between 3 and C − 1, C ≥ 4, and the request graph does not contain a perfect matching. For these open cases, we provide upper bounds that differ from the optimal value by at most one.Peer ReviewedPostprint (published version