531 research outputs found
Disconnected cuts in claw-free graphs.
A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected
subgraph. The corresponding decision problem is called Disconnected Cut. It is known that
Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for
several graph classes. However, the complexity of the problem on claw-free graphs remained an
open question. Its connection to the complexity of the problem to contract a claw-free graph to
the 4-vertex cycle C4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We
prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the
question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of
diameter 2, which we believe is of independent interest and builds on the research line initiated by
Chudnovsky and Seymour (JCTB 2007–2012) and Hermelin et al. (ICALP 2011). On our way to
exploit this decomposition theorem, we characterize how disconnected cuts interact with certain
cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected
Cut is polynomial-time solvable on circular-arc graphs and line graphs
Hamilton cycles in 5-connected line graphs
A conjecture of Carsten Thomassen states that every 4-connected line graph is
hamiltonian. It is known that the conjecture is true for 7-connected line
graphs. We improve this by showing that any 5-connected line graph of minimum
degree at least 6 is hamiltonian. The result extends to claw-free graphs and to
Hamilton-connectedness
The complexity of the Perfect Matching-Cut problem
Perfect Matching-Cut is the problem of deciding whether a graph has a perfect
matching that contains an edge-cut. We show that this problem is NP-complete
for planar graphs with maximum degree four, for planar graphs with girth five,
for bipartite five-regular graphs, for graphs of diameter three and for
bipartite graphs of diameter four. We show that there exist polynomial time
algorithms for the following classes of graphs: claw-free, -free, diameter
two, bipartite with diameter three and graphs with bounded tree-width
The Perfect Matching Cut Problem Revisited
Under embargo until: 2022-09-20In a graph, a perfect matching cut is an edge cut that is a perfect matching. perfect matching cut (pmc) is the problem of deciding whether a given graph has a perfect matching cut, and is known to be NP -complete. We revisit the problem and show that pmc remains NP -complete when restricted to bipartite graphs of maximum degree 3 and arbitrarily large girth. Complementing this hardness result, we give two graph classes in which pmc is polynomial time solvable. The first one includes claw-free graphs and graphs without an induced path on five vertices, the second one properly contains all chordal graphs. Assuming the Exponential Time Hypothesis, we show there is no O∗(2o(n)) -time algorithm for pmc even when restricted to n-vertex bipartite graphs, and also show that pmc can be solved in O∗(1.2721n) time by means of an exact branching algorithm.acceptedVersio
Finding Matching Cuts in H-Free Graphs
The well-known NP-complete problem Matching Cut is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for Matching Cut restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of Matching Cut, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r > 0 such that the first variant is NP-complete for P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs
Hamiltonian chordal graphs are not cycle extendible
In 1990, Hendry conjectured that every Hamiltonian chordal graph is cycle
extendible; that is, the vertices of any non-Hamiltonian cycle are contained in
a cycle of length one greater. We disprove this conjecture by constructing
counterexamples on vertices for any . Furthermore, we show that
there exist counterexamples where the ratio of the length of a non-extendible
cycle to the total number of vertices can be made arbitrarily small. We then
consider cycle extendibility in Hamiltonian chordal graphs where certain
induced subgraphs are forbidden, notably and the bull.Comment: Some results from Section 3 were incorrect and have been removed. To
appear in SIAM Journal on Discrete Mathematic
The Complexity of Surjective Homomorphism Problems -- a Survey
We survey known results about the complexity of surjective homomorphism
problems, studied in the context of related problems in the literature such as
list homomorphism, retraction and compaction. In comparison with these
problems, surjective homomorphism problems seem to be harder to classify and we
examine especially three concrete problems that have arisen from the
literature, two of which remain of open complexity
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