583 research outputs found
Constrained Representations of Map Graphs and Half-Squares
The square of a graph H, denoted H^2, is obtained from H by adding new edges between two distinct vertices whenever their distance in H is two. The half-squares of a bipartite graph B=(X,Y,E_B) are the subgraphs of B^2 induced by the color classes X and Y, B^2[X] and B^2[Y]. For a given graph G=(V,E_G), if G=B^2[V] for some bipartite graph B=(V,W,E_B), then B is a representation of G and W is the set of points in B. If in addition B is planar, then G is also called a map graph and B is a witness of G [Chen, Grigni, Papadimitriou. Map graphs. J. ACM49 (2) (2002) 127-138].
While Chen, Grigni, Papadimitriou proved that any map graph G=(V,E_G) has a witness with at most 3|V|-6 points, we show that, given a map graph G and an integer k, deciding if G admits a witness with at most k points is NP-complete. As a by-product, we obtain NP-completeness of edge clique partition on planar graphs; until this present paper, the complexity status of edge clique partition for planar graphs was previously unknown.
We also consider half-squares of tree-convex bipartite graphs and prove the following complexity dichotomy: Given a graph G=(V,E_G) and an integer k, deciding if G=B^2[V] for some tree-convex bipartite graph B=(V,W,E_B) with |W|<=k points is NP-complete if G is non-chordal dually chordal and solvable in linear time otherwise. Our proof relies on a characterization of half-squares of tree-convex bipartite graphs, saying that these are precisely the chordal and dually chordal graphs
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
Computing Optimal Leaf Roots of Chordal Cographs in Linear Time
A graph G is a k-leaf power, for an integer k >= 2, if there is a tree T with
leaf set V(G) such that, for all vertices x, y in V(G), the edge xy exists in G
if and only if the distance between x and y in T is at most k. Such a tree T is
called a k-leaf root of G. The computational problem of constructing a k-leaf
root for a given graph G and an integer k, if any, is motivated by the
challenge from computational biology to reconstruct phylogenetic trees. For
fixed k, Lafond [SODA 2022] recently solved this problem in polynomial time.
In this paper, we propose to study optimal leaf roots of graphs G, that is,
the k-leaf roots of G with minimum k value. Thus, all k'-leaf roots of G
satisfy k <= k'. In terms of computational biology, seeking optimal leaf roots
is more justified as they yield more probable phylogenetic trees. Lafond's
result does not imply polynomial-time computability of optimal leaf roots,
because, even for optimal k-leaf roots, k may (exponentially) depend on the
size of G. This paper presents a linear-time construction of optimal leaf roots
for chordal cographs (also known as trivially perfect graphs). Additionally, it
highlights the importance of the parity of the parameter k and provides a
deeper insight into the differences between optimal k-leaf roots of even versus
odd k.
Keywords: k-leaf power, k-leaf root, optimal k-leaf root, trivially perfect
leaf power, chordal cographComment: 22 pages, 2 figures, full version of the FCT 2023 pape
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