Graph Treewidth and Geometric Thickness Parameters

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 $\lceil{k/2}\rceil$. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth $k$, the maximum book thickness equals $k$ if $k \leq 2$ and equals $k+1$ if $k \geq 3$. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.Comment: A preliminary version of this paper appeared in the "Proceedings of the 13th International Symposium on Graph Drawing" (GD '05), Lecture Notes in Computer Science 3843:129-140, Springer, 2006. The full version was published in Discrete & Computational Geometry 37(4):641-670, 2007. That version contained a false conjecture, which is corrected on page 26 of this versio

Stack and Queue Layouts via Layered Separators

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph families are not closed under taking minors. For fixed $g$ and $k$, we show that every $n$-vertex graph that can be embedded on a surface of genus $g$ with at most $k$ crossings per edge has stack-number $\mathcal{O}(\log n)$; this includes $k$-planar graphs. The previously best known bound for the stack-number of these families was $\mathcal{O}(\sqrt{n})$, except in the case of $1$-planar graphs. Analogous results are proved for map graphs that can be embedded on a surface of fixed genus. None of these families is closed under taking minors. The main ingredient in the proof of these results is a construction proving that $n$-vertex graphs that admit constant layered separators have $\mathcal{O}(\log n)$ stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Musubime to karamime no gurafu hyogen

制度:新 ; 報告番号:乙2277号 ; 学位の種類:博士(理学) ; 授与年月日:2010/6/22 ; 早大学位記番号:新540

Embedding Schemes for Interconnection Networks.

Graph embeddings play an important role in interconnection network and VLSI design. Designing efficient embedding strategies for simulating one network by another and determining the number of layers required to build a VLSI chip are just two of the many areas in which graph embeddings are used. In the area of network simulation we develop efficient, small dilation embeddings of a butterfly network into a different size and/or type of butterfly network. The genus of a graph gives an indication of how many layers are required to build a circuit. We have determined the exact genus for the permutation network called the star graph, and have given a lower bound for the genus of the permutation network called the pancake graph. The star graph has been proposed as an alternative to the binary hypercube and, therefore, we compare the genus of the star graph with that of the binary hypercube. Another type of embedding that is helpful in determining the number of layers is a book embedding. We develop upper and lower bounds on the pagenumber of a book embedding of the k-ary hypercube along with an upper bound on the cumulative pagewidth

On Compact Encoding of Pagenumber kk Graphs

In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;