11,758 research outputs found
On 3-Coloring Circle Graphs
Given a graph with a fixed vertex order , one obtains a circle
graph whose vertices are the edges of and where two such edges are
adjacent if and only if their endpoints are pairwise distinct and alternate in
. Therefore, the problem of determining whether has a -page book
embedding with spine order is equivalent to deciding whether can be
colored with colors. Finding a -coloring for a circle graph is known to
be NP-complete for and trivial for . For , Unger
(1992) claims an efficient algorithm that finds a 3-coloring in
time, if it exists. Given a circle graph , Unger's algorithm (1) constructs
a 3-\textsc{Sat} formula that is satisfiable if and only if admits a
3-coloring and (2) solves by a backtracking strategy that relies on the
structure imposed by the circle graph. However, the extended abstract misses
several details and Unger refers to his PhD thesis (in German) for details. In
this paper we argue that Unger's algorithm for 3-coloring circle graphs is not
correct and that 3-coloring circle graphs should be considered as an open
problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting
a circle graph whose formula is satisfiable but that is not 3-colorable.
We further show that Unger's backtracking strategy for solving in step
(2) may produce incorrect results and give empirical evidence that it exhibits
a runtime behaviour that is not consistent with the claimed running time.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
Efficient and Perfect domination on circular-arc graphs
Given a graph , a \emph{perfect dominating set} is a subset of
vertices such that each vertex is
dominated by exactly one vertex . An \emph{efficient dominating set}
is a perfect dominating set where is also an independent set. These
problems are usually posed in terms of edges instead of vertices. Both
problems, either for the vertex or edge variant, remains NP-Hard, even when
restricted to certain graphs families. We study both variants of the problems
for the circular-arc graphs, and show efficient algorithms for all of them
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