9 research outputs found
Covering minimal separators and potential maximal cliques in -free graphs
A graph is called -free} if it does not contain a -vertex path as an
induced subgraph. While -free graphs are exactly cographs, the structure
of -free graphs for remains little understood. On one hand,
classic computational problems such as Maximum Weight Independent Set (MWIS)
and -Coloring are not known to be NP-hard on -free graphs for any fixed
. On the other hand, despite significant effort, polynomial-time algorithms
for MWIS in -free graphs~[SODA 2019] and -Coloring in -free
graphs~[Combinatorica 2018] have been found only recently. In both cases, the
algorithms rely on deep structural insights into the considered graph classes.
One of the main tools in the algorithms for MWIS in -free graphs~[SODA
2014] and in -free graphs~[SODA 2019] is the so-called Separator Covering
Lemma that asserts that every minimal separator in the graph can be covered by
the union of neighborhoods of a constant number of vertices. In this note we
show that such a statement generalizes to -free graphs and is false in
-free graphs. We also discuss analogues of such a statement for covering
potential maximal cliques with unions of neighborhoods
Maximum Independent Set when excluding an induced minor: and
Dallard, Milani\v{c}, and \v{S}torgel [arXiv '22] ask if for every class
excluding a fixed planar graph as an induced minor, Maximum Independent Set
can be solved in polynomial time, and show that this is indeed the case when
is any planar complete bipartite graph, or the 5-vertex clique minus one
edge, or minus two disjoint edges. A positive answer would constitute a
far-reaching generalization of the state-of-the-art, when we currently do not
know if a polynomial-time algorithm exists when is the 7-vertex path.
Relaxing tractability to the existence of a quasipolynomial-time algorithm, we
know substantially more. Indeed, quasipolynomial-time algorithms were recently
obtained for the -vertex cycle, [Gartland et al., STOC '21] and the
disjoint union of triangles, [Bonamy et al., SODA '23].
We give, for every integer , a polynomial-time algorithm running in
when is the friendship graph ( disjoint edges
plus a vertex fully adjacent to them), and a quasipolynomial-time algorithm
running in when is (the
disjoint union of triangles and a 4-vertex cycle). The former extends a
classical result on graphs excluding as an induced subgraph [Alekseev,
DAM '07], while the latter extends Bonamy et al.'s result.Comment: 15 pages, 2 figure
Graphs with polynomially many minimal separators
We show that graphs that do not contain a theta, pyramid, prism, or turtle as
an induced subgraph have polynomially many minimal separators. This result is
the best possible in the sense that there are graphs with exponentially many
minimal separators if only three of the four induced subgraphs are excluded. As
a consequence, there is a polynomial time algorithm to solve the maximum weight
independent set problem for the class of (theta, pyramid, prism, turtle)-free
graphs. Since every prism, theta, and turtle contains an even hole, this also
implies a polynomial time algorithm to solve the maximum weight independent set
problem for the class of (pyramid, even hole)-free graphs
Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time
We show that the \textsc{Maximum Weight Independent Set} problem
(\textsc{MWIS}) can be solved in quasi-polynomial time on -free graphs
(graphs excluding a fixed graph as an induced subgraph) for every whose
every connected component is a path or a subdivided claw (i.e., a tree with at
most three leaves). This completes the dichotomy of the complexity of
\textsc{MWIS} in -free graphs for any finite set of
graphs into NP-hard cases and cases solvable in quasi-polynomial time, and
corroborates the conjecture that the cases not known to be NP-hard are actually
polynomial-time solvable.
The key graph-theoretic ingredient in our result is as follows. Fix an
integer . Let be the graph created from three paths on
edges by identifying one endpoint of each path into a single vertex. We
show that, given a graph , one can in polynomial time find either an induced
in , or a balanced separator consisting of \Oh(\log |V(G)|)
vertex neighborhoods in , or an extended strip decomposition of (a
decomposition almost as useful for recursion for \textsc{MWIS} as a partition
into connected components) with each particle of weight multiplicatively
smaller than the weight of . This is a strengthening of a result of Majewski
et al.\ [ICALP~2022] which provided such an extended strip decomposition only
after the deletion of \Oh(\log |V(G)|) vertex neighborhoods. To reach the
final result, we employ an involved branching strategy that relies on the
structural lemma presented above.Comment: 58 pages, 4 figure
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Feedback Vertex Set and Even Cycle Transversal for H-free graphs: Finding large block graphs
We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that for every s \geq 1, both problems are polynomial-time solvable for sP3-free graphs and (sP1 + P5)-free graphs; here, the graph sP3 denotes the disjoint union of s paths on three vertices and the graph sP1 + P5 denotes the disjoint union of s isolated vertices and a path on five vertices. Our new results for Feedback Vertex Set extend all known polynomial-time results for Feedback Vertex Set on H-free graphs, namely for sP2-free graphs [Chiarelli et al., Theoret. Comput. Sci., 705 (2018), pp. 75--83], (sP1 +P3)-free graphs [Dabrowski et al., Algorithmica, 82 (2020), pp. 2841--2866] and P5-free graphs [Abrishami et al., Induced subgraphs of bounded treewidth and the container method, in Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadelphia, 2021, pp. 1948--1964]. Together, the new results also show that both problems exhibit the same behavior on H-free graphs (subject to some open cases). This is in part due to a new general algorithm we design for finding in a (sP3)-free or (sP1 + P5)-free graph G a largest induced subgraph whose blocks belong to some finite class \scrC of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs