777 research outputs found
Determining Finite Connected Graphs Along the Quadratic Embedding Constants of Paths
The QE constant of a finite connected graph , denoted by
, is by definition the maximum of the quadratic function
associated to the distance matrix on a certain sphere of codimension two. We
prove that the QE constants of paths form a strictly increasing sequence
converging to . Then we formulate the problem of determining all the
graphs satisfying
. The answer is
given for and by exploiting forbidden subgraphs for
and the explicit QE constants of star products of the
complete graphs.Comment: 24 pages, 6 figure
b-Coloring Parameterized by Clique-Width
We provide a polynomial-time algorithm for b-Coloring on graphs of constant clique-width. This unifies and extends nearly all previously known polynomial-time results on graph classes, and answers open questions posed by Campos and Silva [Algorithmica, 2018] and Bonomo et al. [Graphs Combin., 2009]. This constitutes the first result concerning structural parameterizations of this problem. We show that the problem is FPT when parameterized by the vertex cover number on general graphs, and on chordal graphs when parameterized by the number of colors. Additionally, we observe that our algorithm for graphs of bounded clique-width can be adapted to solve the Fall Coloring problem within the same runtime bound. The running times of the clique-width based algorithms for b-Coloring and Fall Coloring are tight under the Exponential Time Hypothesis
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