NP-complete problems cannot have efficient algorithms unless P = NP. Due to their importance in practice, however, it is useful to improve the known exponential-time algorithms for NP-complete problems. We survey some of the recent results on such improved exponential-time algorithms for the NP-complete problems satisfiability, graph colorability, and the domatic number problem. The determinist...

The cardinality of a maximum minimal dominating set of a graph is called its upper domination number. The problem of computing this number is generally NP-hard but can be solved in polynomial time in some restricted graph classes. In this work, we consider the complexity and approximability of the weighted version of the problem in two special graph classes: planar bipartite, split. We also pro...

A path in an edge-colored graph is rainbow if no two edges of it are colored the same, and rainbow-connected there a between each pair its vertices. The minimum number colors needed to rainbow-connect G connection G, denoted by rc(G). simple way color spanning tree with distinct then re-use any these remaining G. This proves that rc(G)≤|V(G)|−1. We ask whether stronger tree-like structures colo...

We investigate experimentally the Domatic Partition (DP) problem, the Independent Domatic Partition (IDP) problem and the Idomatic partition problem in Random Geometric Graphs (RGGs). In particular, we model these problems as Integer Linear Programs (ILPs), solve them optimally, and show on a large set of samples that RGGs are independent domatically full most likely (over 93% of the cases) and...