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Edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs

Dettlaff M., Raczek J., Yero I. G.

Opuscula Mathematica

2016

Vol. 36, no. 5

575--588

Given a graph G = (V, E), the subdivision of an edge e = uv ∈ E(G) means the substitution of the edge e by a vertex x and the new edges ux and xv. The domination subdivision number of a graph G is the minimum number of edges of G which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of G is the minimum number of subdivisions which must be done in one edge such that the domination number increases. Moreover, the concepts of paired domination and independent domination subdivision (respectively multisubdivision) numbers are denned similarly. In this paper we study the domination, paired domination and independent domination (subdivision and multisubdivision) numbers of the generalized corona graphs.

Equitable Coloring of Graphs. Recent Theoretical Results and New Practical Algorithms

Furmańczyk H., Jastrzębski A., Kubale M.

Archives of Control Sciences

2016

Vol. 26, no. 3

281--295

In many applications in sequencing and scheduling it is desirable to have an underlaying graph as equitably colored as possible. In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NP-hard, we report on practical experiments with some efficient polynomial-time algorithms for approximate equitable coloring of general graphs.