Rectangle Visibility Numbers of Graphs

Very-Large Scale Integration (VLSI) is the problem of arranging components on the surface of a circuit board and developing the wired network between components. One methodology in VLSI is to treat the entire network as a graph, where the components correspond to vertices and the wired connections correspond to edges. We say that a graph G has a rectangle visibility representation if we can assign each vertex of G to a unique axis-aligned rectangle in the plane such that two vertices u and v are adjacent if and only if there exists an unobstructed horizontal or vertical channel of finite width between the two rectangles that correspond to u and v. If G has such a representation, then we say that G is a rectangle visibility graph. Since it is likely that multiple components on a circuit board may represent the same electrical node, we may consider implementing this idea with rectangle visibility graphs. The rectangle visibility number of a graph G, denoted r(G), is the minimum k such that G has a rectangle visibility representation in which each vertex of G corresponds to at most k rectangles. In this thesis, we prove results on rectangle visibility numbers of trees, complete graphs, complete bipartite graphs, and (1,n)-hilly graphs, which are graphs where there is no path of length 1 between vertices of degree n or more

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

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Three Existence Problems in Extremal Graph Theory

Proving the existence or nonexistence of structures with specified properties is the impetus for many classical results in discrete mathematics. In this thesis we take this approach to three different structural questions rooted in extremal graph theory. When studying graph representations, we seek efficient ways to encode the structure of a graph. For example, an {\it interval representation} of a graph $G$ is an assignment of intervals on the real line to the vertices of $G$ such that two vertices are adjacent if and only if their intervals intersect. We consider graphs that have {\it bar $k$-visibility representations}, a generalization of both interval representations and another well-studied class of representations known as visibility representations. We obtain results on $\mathcal{F}_k$, the family of graphs having bar $k$-visibility representations. We also study $\bigcup_{k=0}^{\infty} \mathcal{F}_k$. In particular, we determine the largest complete graph having a bar $k$-visibility representation, and we show that there are graphs that do not have bar $k$-visibility representations for any $k$. Graphs arise naturally as models of networks, and there has been much study of the movement of information or resources in graphs. Lampert and Slater \cite{LS} introduced {\it acquisition} in weighted graphs, whereby weight moves around $G$ provided that each move transfers weight from a vertex to a heavier neighbor. Our goal in making acquisition moves is to consolidate all of the weight in $G$ on the minimum number of vertices; this minimum number is the {\it acquisition number} of $G$. We study three variations of acquisition in graphs: when a move must transfer all the weight from a vertex to its neighbor, when each move transfers a single unit of weight, and when a move can transfer any positive amount of weight. We consider acquisition numbers in various families of graphs, including paths, cycles, trees, and graphs with diameter $2$. We also study, under the various acquisition models, those graphs in which all the weight can be moved to a single vertex. Restrictive local conditions often have far-reaching impacts on the global structure of mathematical objects. Some local conditions are so limiting that very few objects satisfy the requirements. For example, suppose that we seek a graph in which every two vertices have exactly one common neighbor. Such graphs are called {\it friendship graphs}, and Wilf~\cite{Wilf} proved that the only such graphs consist of edge-disjoint triangles sharing a common vertex. We study a related structural restriction where similar phenomena occur. For a fixed graph $H$, we consider those graphs that do not contain $H$ and such that the addition of any edge completes exactly one copy of $H$. Such a graph is called {\it uniquely $H$-saturated}. We study the existence of uniquely $H$-saturated graphs when $H$ is a path or a cycle. In particular, we determine all of the uniquely $C_4$-saturated graphs; there are exactly ten. Interestingly, the uniquely $C_{5}$-saturated graphs are precisely the friendship graphs characterized by Wilf

Twin-width VIII: delineation and win-wins

We introduce the notion of delineation. A graph class $\mathcal C$ is said delineated if for every hereditary closure $\mathcal D$ of a subclass of $\mathcal C$, it holds that $\mathcal D$ has bounded twin-width if and only if $\mathcal D$ is monadically dependent. An effective strengthening of delineation for a class $\mathcal C$ implies that tractable FO model checking on $\mathcal C$ is perfectly understood: On hereditary closures $\mathcal D$ of subclasses of $\mathcal C$, FO model checking is fixed-parameter tractable (FPT) exactly when $\mathcal D$ has bounded twin-width. Ordered graphs [BGOdMSTT, STOC '22] and permutation graphs [BKTW, JACM '22] 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 show that segment graphs, directed path graphs, 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 '21]. 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 $H_4$-free two-lengthed segment graphs have unbounded twin-width. Our new results, combined with the known FPT algorithm for FO model checking on graphs given with $O(1)$-sequences, lead to win-win arguments. For instance, we derive FPT algorithms for $k$-Ladder on visibility graphs of 1.5D terrains, and $k$-Independent Set on visibility graphs of simple polygons.Comment: 51 pages, 19 figure