7 research outputs found
Multigraph decomposition into stars and into multistars
AbstractWe study the decomposition of multigraphs with a constant edge multiplicity into copies of a fixed star H=K1,t: We present necessary and sufficient conditions for such a decomposition to exist where t=2 and prove NP-completeness of the corresponding decision problem for any t⩾3. We also prove NP-completeness when the edge multiplicity function is not restricted either on the input G or on the fixed multistar H
Multigraph decomposition into multigraphs with two underlying edges
Due to some intractability considerations, reasonable formulation of necessary and sufficient conditions for decomposability of a general multigraph G into a fixed connected multigraph H, is probably not feasible if the underlying simple graph of H has three or more edges. We study the case where H consists of two underlying edges. We present necessary and sufficient conditions for H-decomposability of G, which hold when certain size parameters of G lies within some bounds which depends on the multiplicities of the two edges of H. We also show this result to be "tight" in the sense that even a slight deviation of these size parameters from the given bounds results intractability of the corresponding decision problem
A Linear Kernel for Planar Total Dominating Set
A total dominating set of a graph is a subset such
that every vertex in is adjacent to some vertex in . Finding a total
dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on
general graphs when parameterized by the solution size. By the meta-theorem of
Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total
Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how
such a kernel can be effectively constructed, and how to obtain explicit
reduction rules with reasonably small constants. Following the approach of
Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating
Set on planar graphs with at most vertices, where is the size of the
solution. This result complements several known constructive linear kernels on
planar graphs for other domination problems such as Dominating Set, Edge
Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue
Dominating Set.Comment: 33 pages, 13 figure