1,184 research outputs found
Make a graph singly connected by edge orientations
A directed graph is singly connected if for every ordered pair of
vertices , there is at most one path from to in . Graph
orientation problems ask, given an undirected graph , to find an orientation
of the edges such that the resultant directed graph has a certain property.
In this work, we study the graph orientation problem where the desired property
is that is singly connected. Our main result concerns graphs of a fixed
girth and coloring number . For every , the problem
restricted to instances of girth and coloring number , is either
NP-complete or in P. As further algorithmic results, we show that the problem
is NP-hard on planar graphs and polynomial time solvable distance-hereditary
graphs
The Complexity of Packing Edge-Disjoint Paths
We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem.
Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard.
Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs
Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs
In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
, the class of -graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of . Recently, quite a lot of research has
been devoted to understanding the tractability border for various computational
problems, such as recognition or isomorphism testing, in classes of -graphs
for different graphs . In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree , a polynomial-time algorithm recognizing
-graphs. Tucker showed a polynomial time algorithm recognizing -graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of -graphs is -hard if contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the -hard and
cases of -graphs recognition. First, we show that recognition of
-graphs is -hard when contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing -graphs, where is
a graph containing a cycle and an edge attached to it (-graphs are called
lollipop graphs). Our work leaves open the recognition problems of -graphs
for every unicyclic graph different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of -graphs, where is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of -graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of -graphs, where runs over all unicyclic
graphs, is -complete
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