26 research outputs found
Recognizing Geometric Intersection Graphs Stabbed by a Line
In this paper, we determine the computational complexity of recognizing two
graph classes, \emph{grounded L}-graphs and \emph{stabbable grid intersection}
graphs. An L-shape is made by joining the bottom end-point of a vertical
() segment to the left end-point of a horizontal () segment. The top
end-point of the vertical segment is known as the {\em anchor} of the L-shape.
Grounded L-graphs are the intersection graphs of L-shapes such that all the
L-shapes' anchors lie on the same horizontal line. We show that recognizing
grounded L-graphs is NP-complete. This answers an open question asked by
Jel{\'\i}nek \& T{\"o}pfer (Electron. J. Comb., 2019).
Grid intersection graphs are the intersection graphs of axis-parallel line
segments in which two vertical (similarly, two horizontal) segments cannot
intersect. We say that a (not necessarily axis-parallel) straight line
stabs a segment , if intersects . A graph is a stabbable grid
intersection graph () if there is a grid intersection representation
of in which the same line stabs all its segments. We show that recognizing
graphs is -complete, even on a restricted class of graphs. This
answers an open question asked by Chaplick \etal (\textsc{O}rder, 2018).Comment: 18 pages, 11 Figure
Twin-Width VIII: Delineation and Win-Wins
We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated) if for every hereditary closure D of a subclass of C, it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures of subclasses D of C, FO model checking on D is fixed-parameter tractable (FPT) exactly when D has bounded twin-width. Ordered graphs [BGOdMSTT, STOC \u2722] and permutation graphs [BKTW, JACM \u2722] are effectively delineated, while subcubic graphs are not. On the one hand, we prove that interval graphs, and even, rooted directed path graphs are delineated. On the other hand, we observe or show that segment graphs, directed path graphs (with arbitrarily many roots), and visibility graphs of simple polygons are not delineated.
In an effort to draw the delineation frontier between interval graphs (that are delineated) and axis-parallel two-lengthed segment graphs (that are not), we investigate the twin-width of restricted segment intersection classes. It was known that (triangle-free) pure axis-parallel unit segment graphs have unbounded twin-width [BGKTW, SODA \u2721]. We show that K_{t,t}-free segment graphs, and axis-parallel H_t-free unit segment graphs have bounded twin-width, where H_t is the half-graph or ladder of height t. In contrast, axis-parallel H?-free two-lengthed segment graphs have unbounded twin-width. We leave as an open question whether unit segment graphs are delineated.
More broadly, we explore which structures (large bicliques, half-graphs, or independent sets) are responsible for making the twin-width large on the main classes of intersection and visibility graphs. Our new results, combined with the FPT algorithm for first-order model checking on graphs given with O(1)-sequences [BKTW, JACM \u2722], give rise to a variety of algorithmic win-win arguments. They all fall in the same framework: If p is an FO definable graph parameter that effectively functionally upperbounds twin-width on a class C, then p(G) ? k can be decided in FPT time f(k) ? |V(G)|^O(1). For instance, we readily derive FPT algorithms for k-Ladder on visibility graphs of 1.5D terrains, and k-Independent Set on visibility graphs of simple polygons. This showcases that the theory of twin-width can serve outside of classes of bounded twin-width
Kempe Equivalent List Colorings Revisited
A classical theorem of Gallai states that in every graph that is critical for
-colorings, the vertices of degree induce a tree-like graph whose
blocks are either complete graphs or cycles of odd length. Borodin and,
independently, Erd\H{o}s et al. provided a well-known generalization of
Gallai's Theorem to list colorings, where the list at each vertex has the same
number of available colors as the degree of that vertex. In this paper, we
obtain an analogous result for Kempe equivalence of list colorings, partially
resolving a problem of Cranston and Mahmoud.Comment: 8 pages; strengthened main result and improved expositio
Isometric path complexity of graphs
A set of isometric paths of a graph is "-rooted", where is a
vertex of , if is one of the end-vertices of all the isometric paths in
. The isometric path complexity of a graph , denoted by , is the
minimum integer such that there exists a vertex satisfying the
following property: the vertices of any isometric path of can be
covered by many -rooted isometric paths.
First, we provide an -time algorithm to compute the isometric path
complexity of a graph with vertices and edges. Then we show that the
isometric path complexity remains bounded for graphs in three seemingly
unrelated graph classes, namely, hyperbolic graphs, (theta, prism,
pyramid)-free graphs, and outerstring graphs. Hyperbolic graphs are extensively
studied in Metric Graph Theory. The class of (theta, prism, pyramid)-free
graphs are extensively studied in Structural Graph Theory, e.g. in the context
of the Strong Perfect Graph Theorem. The class of outerstring graphs is studied
in Geometric Graph Theory and Computational Geometry. Our results also show
that the distance functions of these (structurally) different graph classes are
more similar than previously thought.
There is a direct algorithmic consequence of having small isometric path
complexity. Specifically, we show that if the isometric path complexity of a
graph is bounded by a constant, then there exists a polynomial-time
constant-factor approximation algorithm for ISOMETRIC PATH COVER, whose
objective is to cover all vertices of a graph with a minimum number of
isometric paths. This applies to all the above graph classes.Comment: A preliminary version appeared in the proceedings of the MFCS 2023
conferenc
Algorithms and Complexity for Geodetic Sets on Planar and Chordal Graphs
We study the complexity of finding the geodetic number on subclasses of planar graphs and chordal graphs. A set S of vertices of a graph G is a geodetic set if every vertex of G lies in a shortest path between some pair of vertices of S. The Minimum Geodetic Set (MGS) problem is to find a geodetic set with minimum cardinality of a given graph. The problem is known to remain NP-hard on bipartite graphs, chordal graphs, planar graphs and subcubic graphs. We first study MGS on restricted classes of planar graphs: we design a linear-time algorithm for MGS on solid grids, improving on a 3-approximation algorithm by Chakraborty et al. (CALDAM, 2020) and show that MGS remains NP-hard even for subcubic partial grids of arbitrary girth. This unifies some results in the literature. We then turn our attention to chordal graphs, showing that MGS is fixed parameter tractable for inputs of this class when parameterized by their treewidth (which equals the clique number minus one). This implies a linear-time algorithm for k-trees, for fixed k. Then, we show that MGS is NP-hard on interval graphs, thereby answering a question of Ekim et al. (LATIN, 2012). As interval graphs are very constrained, to prove the latter result we design a rather sophisticated reduction technique to work around their inherent linear structure