94,964 research outputs found
Long induced paths in graphs
We prove that every 3-connected planar graph on vertices contains an
induced path on vertices, which is best possible and improves
the best known lower bound by a multiplicative factor of . We
deduce that any planar graph (or more generally, any graph embeddable on a
fixed surface) with a path on vertices, also contains an induced path on
vertices. We conjecture that for any , there is a
contant such that any -degenerate graph with a path on vertices
also contains an induced path on vertices. We provide
examples showing that this order of magnitude would be best possible (already
for chordal graphs), and prove the conjecture in the case of interval graphs.Comment: 20 pages, 5 figures - revised versio
Complexity results for matching cut problems in graphs without long induced paths
In a graph, a (perfect) matching cut is an edge cut that is a (perfect)
matching. Matching Cut (MC), respectively, Perfect Matching Cut (PMC), is the
problem of deciding whether a given graph has a matching cut, respectively, a
perfect matching cut. The Disconnected Perfect Matching problem (DPM) is to
decide if a graph has a perfect matching that contains a matching cut. Solving
an open problem recently posed in [Lucke, Paulusma, Ries (ISAAC 2022), and
Feghali, Lucke, Paulusma, Ries (arXiv:2212.12317)], we show that PMC is
NP-complete in graphs without induced 14-vertex path . Our reduction
also works simultaneously for MC and DPM, improving the previous hardness
results of MC on -free graphs and of DPM on -free graphs to
-free graphs for both problems.
Actually, we prove a slightly stronger result: within -free graphs,
it is hard to distinguish between (i) those without matching cuts and those in
which every matching cut is a perfect matching cut, (ii) those without perfect
matching cuts and those in which every matching cut is a perfect matching cut,
and (iii) those without disconnected perfect matchings and those in which every
matching cut is a perfect matching cut.
Moreover, assuming the Exponential Time Hypothesis, none of these problems
can be solved in time for -vertex -free input graphs.
We also consider the problems in graphs without long induced cycles. It is
known that MC is polynomially solvable in graphs without induced cycles of
length at least 5 [Moshi (JGT 1989)]. We point out that the same holds for DPM.Comment: To appear in the proceedings of WG 202
Maximum Independent Sets in Subcubic Graphs: New Results
International audienceWe consider the complexity of the classical Independent Set problem on classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. It is well-known that a necessary condition for Independent Set to be tractable in such a class (unless P=NP) is that the set of forbidden induced subgraphs includes a subdivided star S k,k,k , for some k. Here, S k,k,k is the graph obtained by taking three paths of length k and identifying one of their endpoints. It is an interesting open question whether this condition is also sufficient: is Independent Set tractable on all hereditary classes of subcu-bic graphs that exclude some S k,k,k ? A positive answer to this question would provide a complete classification of the complexity of Independent Set on all classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. The best currently known result of this type is tractability for S2,2,2-free graphs. In this paper we generalize this result by showing that the problem remains tractable on S 2,k,k-free graphs, for any fixed k. Along the way, we show that subcubic Independent Set is tractable for graphs excluding a type of graph we call an "apple with a long stem", generalizing known results for apple-free graphs
Long Circuits and Large Euler Subgraphs
An undirected graph is Eulerian if it is connected and all its vertices are
of even degree. Similarly, a directed graph is Eulerian, if for each vertex its
in-degree is equal to its out-degree. It is well known that Eulerian graphs can
be recognized in polynomial time while the problems of finding a maximum
Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this
paper, we study the parameterized complexity of the following Euler subgraph
problems:
- Large Euler Subgraph: For a given graph G and integer parameter k, does G
contain an induced Eulerian subgraph with at least k vertices?
- Long Circuit: For a given graph G and integer parameter k, does G contain
an Eulerian subgraph with at least k edges?
Our main algorithmic result is that Large Euler Subgraph is fixed parameter
tractable (FPT) on undirected graphs. We find this a bit surprising because the
problem of finding an induced Eulerian subgraph with exactly k vertices is
known to be W[1]-hard. The complexity of the problem changes drastically on
directed graphs. On directed graphs we obtained the following complexity
dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable
in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT
on directed and undirected graphs
Exploiting structure to cope with NP-hard graph problems: Polynomial and exponential time exact algorithms
An ideal algorithm for solving a particular problem always finds an optimal solution, finds such a solution for every possible instance, and finds it in polynomial time. When dealing with NP-hard problems, algorithms can only be expected to possess at most two out of these three desirable properties. All algorithms presented in this thesis are exact algorithms, which means that they always find an optimal solution. Demanding the solution to be optimal means that other concessions have to be made when designing an exact algorithm for an NP-hard problem: we either have to impose restrictions on the instances of the problem in order to achieve a polynomial time complexity, or we have to abandon the requirement that the worst-case running time has to be polynomial. In some cases, when the problem under consideration remains NP-hard on restricted input, we are even forced to do both.
Most of the problems studied in this thesis deal with partitioning the vertex set of a given graph. In the other problems the task is to find certain types of paths and cycles in graphs. The problems all have in common that they are NP-hard on general graphs. We present several polynomial time algorithms for solving restrictions of these problems to specific graph classes, in particular graphs without long induced paths, chordal graphs and claw-free graphs. For problems that remain NP-hard even on restricted input we present exact exponential time algorithms. In the design of each of our algorithms, structural graph properties have been heavily exploited. Apart from using existing structural results, we prove new structural properties of certain types of graphs in order to obtain our algorithmic results
Algorithms for square-3PC(·, ·)-free Berge graphs
We consider the class of graphs containing no odd hole, no odd antihole, and no configuration consisting of three paths between two nodes such that any two of the paths
induce a hole, and at least two of the paths are of length 2. This class generalizes clawfree Berge graphs and square-free Berge graphs. We give a combinatorial algorithm of
complexity O(n7) to find a clique of maximum weight in such a graph. We also consider several subgraph-detection problems related to this class
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