arXiv subjects
Publications and source records attributed to Philipp Kindermann.
The problem of deciding whether a biconnected planar digraph $G=(V,E)$ can be augmented to become an $st$-planar graph by adding a set of oriented edges $E' \subseteq V \times V$ is known to be NP-complete. We show that the problem is fixed-parameter tractable when parameterized by the size of the set $E'$.
A pair $\langle G_0, G_1 \rangle$ of graphs admits a mutual witness proximity drawing $\langle Γ_0, Γ_1 \rangle$ when: (i) $Γ_i$ represents $G_i$, and (ii) there is an edge $(u,v)$ in $Γ_i$ if and only if there is no vertex $w$ in $Γ_{1-i}$ that is ``too close'' to both $u$ and $v$ ($i=0,1$). In this paper, we consider infinitely many definitions of closeness by adopting the $β$-proximity rule for any $β\in [1,\infty]$ and study pairs of isomorphic trees that admit a mutual witness $β$-proximity drawing. Specifically, we show that every two isomorphic trees admit a mutual witness $β$-proximity drawing for any $β\in [1,\infty]$. The constructive technique can be made ``robust'': For some tree pairs we can suitably prune linearly many leaves from one of the two trees and still retain their mutual witness $β$-proximity drawability. Notably, in the special case of isomorphic caterpillars and $β=1$, we construct linearly separable mutual witness Gabriel drawings.
Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both NP-complete, but XP when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known $n^{O(tw)}$-time algorithms cannot be improved to run in time $n^{o(tw)}$ unless ETH fails.
Bipartite graphs model the relationships between two disjoint sets of entities in several applications and are naturally drawn as 2-layer graph drawings. In such drawings, the two sets of entities (vertices) are placed on two parallel lines (layers), and their relationships (edges) are represented by segments connecting vertices. Methods for constructing 2-layer drawings often try to minimize the number of edge crossings. We use vertex splitting to reduce the number of crossings, by replacing selected vertices on one layer by two (or more) copies and suitably distributing their incident edges among these copies. We study several optimization problems related to vertex splitting, either minimizing the number of crossings or removing all crossings with fewest splits. While we prove that some variants are \NP-complete, we obtain polynomial-time algorithms for others. We run our algorithms on a benchmark set of bipartite graphs representing the relationships between human anatomical structures and cell types.
We continue the study of linear layouts of graphs in relation to known data structures. At a high level, given a data structure, the goal is to find a linear order of the vertices of the graph and a partition of its edges into pages, such that the edges in each page follow the restriction of the given data structure in the underlying order. In this regard, the most notable representatives are the stack and queue layouts, while there exists some work also for deques. In this paper, we study linear layouts of graphs that follow the restriction of a restricted-input queue (rique), in which insertions occur only at the head, and removals occur both at the head and the tail. We characterize the graphs admitting rique layouts with a single page and we use the characterization to derive a corresponding testing algorithm when the input graph is maximal planar. We finally give bounds on the number of needed pages (so-called rique-number) of complete graphs.
A rectangular dual of a plane graph $G$ is a contact representations of $G$ by interior-disjoint axis-aligned rectangles such that (i) no four rectangles share a point and (ii) the union of all rectangles is a rectangle. A rectangular dual gives rise to a regular edge labeling (REL), which captures the orientations of the rectangle contacts. We study the problem of morphing between two rectangular duals of the same plane graph. If we require that, at any time throughout the morph, there is a rectangular dual, then a morph exists only if the two rectangular duals realize the same REL. Therefore, we allow intermediate contact representations of non-rectangular polygons of constant complexity. Given an $n$-vertex plane graph, we show how to compute in $O(n^3)$ time a piecewise linear morph that consists of $O(n^2)$ linear morphing steps.
A graph $G$ is well-covered if every minimal vertex cover of $G$ is minimum, and a graph $G$ is well-dominated if every minimal dominating set of $G$ is minimum. Studies on well-covered graphs were initiated in [Plummer, JCT 1970], and well-dominated graphs were first introduced in [Finbow, Hartnell and Nowakow, AC 1988]. Well-dominated graphs are well-covered, and both classes have been widely studied in the literature. The recognition of well-covered graphs was proved coNP-complete by [Chvátal and Slater, AODM 1993] and by [Sankaranarayana and Stewart, Networks 1992], but the complexity of recognizing well-dominated graphs has been left open since their introduction. We close this complexity gap by proving that recognizing well-dominated graphs is coNP-complete. This solves a well-known open question (c.f. [Levit and Tankus, DM 2017] and [Gözüpek, Hujdurovic and Milanič, DMTCS 2017]), which was first asked in [Caro, Sebő and Tarsi, JAlg 1996]. Surprisingly, our proof is quite simple, although it was a long-standing open problem. Finally, we show that recognizing well-totally-dominated graphs is coNP-complete, answering a question of [Bahad\ir, Ekim, and Gözüpek, AMC 2021].
A rectangular dual of a graph $G$ is a contact representation of $G$ by axis-aligned rectangles such that (i)~no four rectangles share a point and (ii)~the union of all rectangles is a rectangle. The partial representation extension problem for rectangular duals asks whether a given partial rectangular dual can be extended to a rectangular dual, that is, whether there exists a rectangular dual where some vertices are represented by prescribed rectangles. Combinatorially, a rectangular dual can be described by a regular edge labeling (REL), which determines the orientations of the rectangle contacts. We describe two approaches to solve the partial representation extension problem for rectangular duals with given REL. On the one hand, we characterise the RELs that admit an extension, which leads to a linear-time testing algorithm. In the affirmative, we can construct an extension in linear time. This partial representation extension problem can also be formulated as a linear program (LP). We use this LP to solve the simultaneous representation problem for the case of rectangular duals when each input graph is given together with a REL.
We consider the problem of drawing an outerplanar graph with $n$ vertices with at most one bend per edge if the outer face is already drawn as a simple polygon. We prove that it can be decided in $O(nm)$ time if such a drawing exists, where $m\le n-3$ is the number of interior edges. In the positive case, we can also compute such a drawing.
A mixed s-stack q-queue layout of a graph consists of a linear order of its vertices and of a partition of its edges into s stacks and q queues, such that no two edges in the same stack cross and no two edges in the same queue nest. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed 1-stack 1-queue layout. Recently, Pupyrev disproved this conjectured by demonstrating a planar partial 3-tree that does not admit a 1-stack 1-queue layout. In this note, we strengthen Pupyrev's result by showing that the conjecture does not hold even for 2-trees, also known as series-parallel graphs.
A rectilinear polygon is a polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus 4. It is known that any such sequence can be realized by a rectilinear polygon. In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are NP-hard in general. This answers an open question of Patrignani [CGTA 2001], who showed that it is NP-hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing polylines with minimum bounding box area is NP-hard. Then we consider the special cases of $x$-monotone and $xy$-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.
Stick graphs are intersection graphs of horizontal and vertical line segments that all touch a line of slope -1 and lie above this line. De Luca et al. [GD'18] considered the recognition problem of stick graphs when no order is given (STICK), when the order of either one of the two sets is given (STICK_A), and when the order of both sets is given (STICK_AB). They showed how to solve STICK_AB efficiently. In this paper, we improve the running time of their algorithm, and we solve STICK_A efficiently. Further, we consider variants of these problems where the lengths of the sticks are given as input. We show that these variants of STICK, STICK_A, and STICK_AB are all NP-complete. On the positive side, we give an efficient solution for STICK_AB with fixed stick lengths if there are no isolated vertices.
We study the 1-planar, quasi-planar, and fan-planar crossing number in comparison to the (unrestricted) crossing number of graphs. We prove that there are $n$-vertex 1-planar (quasi-planar, fan-planar) graphs such that any 1-planar (quasi-planar, fan-planar) drawing has $Ω(n)$ crossings, while $O(1)$ crossings suffice in a crossing-minimal drawing without restrictions on local edge crossing patterns.
A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and drawing succinctness. We introduce greedy rectilinear drawings, in which each edge is either a horizontal or a vertical segment. These drawings have several properties that improve human readability and support network routing. We address the problem of testing whether a planar rectilinear representation, i.e., a plane graph with specified vertex angles, admits vertex coordinates that define a greedy drawing. We provide a characterization, a linear-time testing algorithm, and a full generative scheme for universal greedy rectilinear representations, i.e., those for which every drawing is greedy. For general greedy rectilinear representations, we give a combinatorial characterization and, based on it, a polynomial-time testing and drawing algorithm for a meaningful subset of instances.
The visual complexity of a graph drawing can be measured by the number of geometric objects used for the representation of its elements. In this paper, we study planar graph drawings where edges are represented by few segments. In such a drawing, one segment may represent multiple edges forming a path. Drawings of planar graphs with few segments were intensively studied in the past years. However, the area requirements were only considered for limited subclasses of planar graphs. In this paper, we show that trees have drawings with $3n/4-1$ segments and $n^2$ area, improving the previous result of $O(n^{3.58})$. We also show that 3-connected planar graphs and biconnected outerplanar graphs have a drawing with $8n/3-O(1)$ and $3n/2-O(1)$ segments, respectively, and $O(n^3)$ area.
Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in $R^3$, so their projections should be smooth curves in $R^2$ with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset $\varepsilon$, while maintaining a $180^\circ$ angle between opposite edges.
It is well-known that every planar graph has a Tutte path, i.e., a path $P$ such that any component of $G-P$ has at most three attachment points on $P$. However, it was only recently shown that such Tutte paths can be found in polynomial time. In this paper, we give a new proof that 3-connected planar graphs have Tutte paths, which leads to a linear-time algorithm to find Tutte paths. Furthermore, our Tutte path has special properties: it visits all exterior vertices, all components of $G-P$ have exactly three attachment points, and we can assign distinct representatives to them that are interior vertices. Finally, our running time bound is slightly stronger; we can bound it in terms of the degrees of the faces that are incident to $P$. This allows us to find some applications of Tutte paths (such as binary spanning trees and 2-walks) in linear time as well.
$Θ_6$-Graphs graphs are important geometric graphs that have many applications especially in wireless sensor networks. They are equivalent to Delaunay graphs where empty equilateral triangles take the place of empty circles. We investigate lower bounds on the size of maximum matchings in these graphs. The best known lower bound is $n/3$, where $n$ is the number of vertices of the graph. Babu et al. (2014) conjectured that any $Θ_6$-graph has a (near-)perfect matching (as is true for standard Delaunay graphs). Although this conjecture remains open, we improve the lower bound to $(3n-8)/7$. We also relate the size of maximum matchings in $Θ_6$-graphs to the minimum size of a blocking set. Every edge of a $Θ_6$-graph on point set $P$ corresponds to an empty triangle that contains the endpoints of the edge but no other point of $P$. A blocking set has at least one point in each such triangle. We prove that the size of a maximum matching is at least $β(n)/2$ where $β(n)$ is the minimum, over all theta-six graphs with $n$ vertices, of the minimum size of a blocking set. In the other direction, lower bounds on matchings can be used to prove bounds on $β$, allowing us to show that $β(n)\geq 3n/4-2$.