arXiv subjects

Van Bang Le

Publications and source records attributed to Van Bang Le.

22 records · Page 2Linked to original sources

arXiv2014

Polynomial time recognition of squares of ptolemaic graphs and 3-sun-free split graphs

The square of a graph $G$, denoted $G^2$, is obtained from $G$ by putting an edge between two distinct vertices whenever their distance is two. Then $G$ is called a square root of $G^2$. Deciding whether a given graph has a square root is known to be NP-complete, even if the root is required to be a chordal graph or even a split graph. We present a polynomial time algorithm that decides whether a given graph has a ptolemaic square root. If such a root exists, our algorithm computes one with a minimum number of edges. In the second part of our paper, we give a characterization of the graphs that admit a 3-sun-free split square root. This characterization yields a polynomial time algorithm to decide whether a given graph has such a root, and if so, to compute one.

cs.DM↗

arXiv2013

Complexity Results for Rainbow Matchings

A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, \mrbm: Given an edge-colored graph $G$, how large is the largest rainbow matching in $G$? We present several sharp contrasts in the complexity of this problem. We show, among others, that * can be approximated by a polynomial algorithm with approximation ratio $2/3-\eps$. * is APX-complete, even when restricted to properly edge-colored linear forests without a $5$-vertex path, and is solvable in %time $O(m^{3/2})$ on edge-colored $m$-edge polynomial time for edge-colored forests without a $4$-vertex path. * is APX-complete, even when restricted to properly edge-colored trees without an $8$-vertex path, and is solvable in %time $O(n^{7/2})$ on edge-colored $n$-vertex polynomial time for edge-colored trees without a $7$-vertex path. * is APX-complete, even when restricted to properly edge-colored paths. These results provide a dichotomy theorem for the complexity of the problem on forests and trees in terms of forbidding paths. The latter is somewhat surprising, since, to the best of our knowledge, no (unweighted) graph problem prior to our result is known to be NP-hard for simple paths. We also address the parameterized complexity of the problem.

cs.DM↗

arXiv2013

Forests and Trees among Gallai Graphs

The Gallai graph $Γ(G)$ of a graph $G$ has the edges of $G$ as its vertices and two distinct vertices $e$ and $f$ of $Γ(G)$ are adjacent in $Γ(G)$ if the edges $e$ and $f$ of $G$ are adjacent in $G$ but do not span a triangle in $G$. Clearly, $Γ(G)$ is a subgraph of the line graph of $G$. While line graphs can be recognized efficiently the complexity of recognizing Gallai graphs is unknown. In the present paper we characterize those graphs whose Gallai graphs are forests or trees, respectively.

math.CO↗

arXiv2009

Computing Graph Roots Without Short Cycles

Graph G is the square of graph H if two vertices x, y have an edge in G if and only if x, y are of distance at most two in H. Given H it is easy to compute its square H2, however Motwani and Sudan proved that it is NP-complete to determine if a given graph G is the square of some graph H (of girth 3). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if G = H2 for some graph H of small girth. The main results are the following. - There is a graph theoretical characterization for graphs that are squares of some graph of girth at least 7. A corollary is that if a graph G has a square root H of girth at least 7 then H is unique up to isomorphism. - There is a polynomial time algorithm to recognize if G = H2 for some graph H of girth at least 6. - It is NP-complete to recognize if G = H2 for some graph H of girth 4. These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

cs.DM↗