1,740 research outputs found
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
Linearizing Partial Search Orders
In recent years, questions about the construction of special orderings of a
given graph search were studied by several authors. On the one hand, the so
called end-vertex problem introduced by Corneil et al. in 2010 asks for search
orderings ending in a special vertex. On the other hand, the problem of finding
orderings that induce a given search tree was introduced already in the 1980s
by Hagerup and received new attention most recently by Beisegel et al. Here, we
introduce a generalization of some of these problems by studying the question
whether there is a search ordering that is a linear extension of a given
partial order on a graph's vertex set. We show that this problem can be solved
in polynomial time on chordal bipartite graphs for LBFS, which also implies the
first polynomial-time algorithms for the end-vertex problem and two search tree
problems for this combination of graph class and search. Furthermore, we
present polynomial-time algorithms for LBFS and MCS on split graphs which
generalize known results for the end-vertex and search tree problems.Comment: full version of an extended abstract to be published in the
Proceedings of the 48th International Workshop on Graph-Theoretic Concepts in
Computer Science (WG 2022) in T\"ubinge
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