Michael B. Dillencourt, David Eppstein, Daniel S. Hirschberg
备注：
11 pages, 3 figures. A preliminary version of this paper appeared in the Sixth Symposium on Graph Drawing, GD '98, (Montréal, Canada, August 1998), Springer-Verlag Lecture Notes in Computer Science 1547, 102--110
发表信息：
J. Graph Algorithms & Applications 4(3):5-17, 2000
We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straight-line edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graph-theoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, K_n. We show that the geometric thickness of K_n lies between ceiling((n/5.646) + 0.342) and ceiling(n/4), and we give exact values of the geometric thickness of K_n for n <= 12 and n in {15,16}. We also consider the geometric thickness of the family of complete bipartite graphs. In particular, we show that, unlike the case of complete graphs, there are complete bipartite graphs with arbitrarily large numbers of vertices for which the geometric thickness coincides with the standard graph-theoretical thickness.
We show that geometric thickness and book thickness are not asymptotically equivalent: for every t, there exists a graph with geometric thickness two and book thickness >= t.
We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness >= t.
The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In: Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists Delta-regular graphs with arbitrarily large geometric thickness. In particular, for all Delta >= 9 and for all large n, there exists a Delta-re…
A graph is 2-degenerate if every subgraph contains a vertex of degree at most 2. We show that every 2-degenerate graph can be drawn with straight lines such that the drawing decomposes into 4 plane forests. Therefore, the geometric arboricity, and hence the geometric thickness, of 2-degenerate graphs is at most 4. On the other hand, we show that there are 2-degenerate graphs that do not admit any straight-line drawing with a decomposition of the edge set into 2 plane graphs. That is, there are 2-degenerate graphs with geometric thickness, and hence geometric arboricity, at least 3. This answer…
Consider a drawing of a graph $G$ in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of $G$, is the classical graph parameter "thickness". By restricting the edges to be straight, we obtain the "geometric thickness". By further restricting the vertices to be in convex position, we obtain the "book thickness". This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth $k$, the maximum thickness and the maximum geometric thickness both equal $\lceil{k/2}\…
The $g$-girth-thickness $θ(g,G)$ of a graph $G$ is the smallest number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we calculate the $4$-girth-thickness $θ(4,G)$ of the complete $m$-partite graph $G$ when each part has an even number of vertices.
In this paper, we define the $4$-girth-thickness $θ(4,G)$ of a graph $G$ as the minimum number of planar subgraphs of girth at least $4$ whose union is $G$. We obtain the $4$-girth-thickness of the arbitrary complete graph $K_n$ getting that $θ(4,K_n)=\left\lceil \frac{n+2}{4}\right\rceil$ for $n\not=6,10$ and $θ(4,K_6)=3$.
The $g$-girth-thickness $θ(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this paper, we determine the $6$-girth-thickness $θ(6,K_n)$ of the complete graph $K_n$ in almost all cases. And also, we calculate by computer the missing value of $θ(4,K_n)$.
The $g$-girth-thickness $θ(g,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least $g$ whose union is $G$. In this note, we give the $4$-girth-thickness $θ(4,L(K_n))$ of the line graph of the complete graph $L(K_n)$ when $n$ is even. We also give the minimum number of subgraphs of $L(K_n)$, which are of girth at least $4$ and embeddable on the projective plane, whose union is $L(K_n)$.
The thickness of a graph G is the minimum number of planar subgraphs whose union is G. In this paper, we obtain the thickness of complete 3-partite graph K_1,n,n, K_2,n,n and complete 4-partite graph K_1,1,n,n.
The thickness $θ(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. It is a topological invariant of a graph, which was defined by W.T. Tutte in 1963 and also has important applications to VLSI design. But comparing with other topological invariants, e.g. genus and crossing number, results about thickness of graphs are few. The only types of graphs whose thicknesses have been obtained are complete graphs, complete bipartite graphs and hypercubes. In this paper, by operations on graphs, the thickness of the cartesian product $K_n \B…
The $4$-girth-thickness $θ(4,G)$ of a graph $G$ is the minimum number of planar subgraphs of girth at least four whose union is $G$. In this paper, we obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is $\big\lceil\frac{n+1}{2}\big\rceil$ except for $θ(4,K_{1,1,1})=2$. And we also show that the $4$-girth-thickness of the complete graph $K_{10}$ is three which disprove the conjecture $θ(4,K_{10})=4$ posed by Rubio-Montiel (Ars Math Contemp 14(2) (2018) 319).
Motivated by applications in graph drawing and information visualization, we examine the planar split thickness of a graph, that is, the smallest $k$ such that the graph is $k$-splittable into a planar graph. A $k$-split operation substitutes a vertex $v$ by at most $k$ new vertices such that each neighbor of $v$ is connected to at least one of the new vertices. We first examine the planar split thickness of complete graphs, complete bipartite graphs, multipartite graphs, bounded degree graphs, and genus-1 graphs. We then prove that it is NP-hard to recognize graphs that are $2$-splittable int…
The $book$ $embedding$ of a graph $G$ is to place the vertices of $G$ on the spine and draw the edges to the pages so that the edges in the same page do not cross with each other. The book embedding is $matching$ if the pages have maximum degree $1$. The $matching$ $book$ $thickness$ is the minimum number of pages in which graphs can be matching book embedded. In this paper, we show that the matching book thickness of the Cartesian product $K_p\Box C_q$of a complete graph $K_p$ and a cycle $C_q$ is equal to $Δ(K_p\Box C_q)+1$.
In response to a well-known open question ``Does every complete geometric graph on $2n\/$ vertices have a partition of its edge set into $n\/$ plane spanning trees?" we provide an affirmative answer when the complete geometry graph is in the regular wheel configuration. Also we present sufficient conditions for the complete geometric graph on $2n\/$ vertices to have a partition of its edge set into $n\/$ plane spanning trees (which are double stars, caterpillars or $ w\/$-caterpillars).
Given a simplicial complex $K$, we consider several notions of geometric complexity of embeddings of $K$ in a Euclidean space ${\mathbb R}^d$: thickness, distortion, and refinement complexity (the minimal number of simplices needed for a PL embedding). We show that any $n$-complex with $N$ simplices which topologically embeds in ${\mathbb R}^{2n}$, $n>2$, can be PL embedded in ${\mathbb R}^{2n}$ with refinement complexity $O(e^{N^{4+ε}})$. Families of simplicial $n$-complexes $K$ are constructed such that any embedding of $K$ into ${\mathbb R}^{2n}$ has an exponential lower bound on thickness …
In this paper we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs, and in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.
Counting the number of Hamiltonian cycles that are contained in a geometric graph is {\bf \#P}-complete even if the graph is known to be planar \cite{lot:refer}. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set $P\/$ of $n\/$ points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph $K_n\/$? We investigate the problem by taking two different situations of $P\/$, namely, when $P\/$ is in convex positio…
We show that the decision problem of determining whether a given (abstract simplicial) $k$-complex has a geometric embedding in $\mathbb R^d$ is complete for the Existential Theory of the Reals for all $d\geq 3$ and $k\in\{d-1,d\}$. This implies that the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution. Moreover, this implies NP-hardness and constitutes the first hardness results for the algorithmic problem of geometric embedding (abstract simplicial) complexes.
Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into beyond planar subgraphs, namely into $k$-planar and $k$-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.
In this paper, we disprove the long-standing conjecture that any complete geometric graph on $2n$ vertices can be partitioned into $n$ plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which \emph{cannot} be partitioned into plane spanning trees (or even into arbitrary plane \emph{subgraphs}), including a complete description of all possible partitions (into plane spanning trees). Furthermore, we show a sufficient condition for \emph{generalized wheels} to not admit a partition into plane spanning trees…
We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.
