Universal Graphs for Bounded-Degree Trees and Planar Graphs
نویسندگان
چکیده
How small can a graph be that contains as subgraphs all trees on n vertices with maximum degree d? In this paper, this question is answered by constructing such universal graphs that have n vertices and bounded degree (depending only on d). Universal graphs with n vertices and O(n log n) edges are also constructed that contain all bounded-degree planar graphs on n vertices as subgraphs. In general, it is shown that the minimum universal graph containing all bounded-degree graphs on n vertices with separators of size n has O(n) edges if a < 1/2; O(n log n) edges if a 1/2; O(n2.) edges if a > 1/2.
منابع مشابه
Labeling Schemes for Bounded Degree Graphs
We investigate adjacency labeling schemes for graphs of bounded degree ∆ = O(1). In particular, we present an optimal (up to an additive constant) log n+ O(1) adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud an...
متن کاملCovering planar graphs with forests
We study the problem of covering graphs with trees and a graph of bounded maximum degree. By a classical theorem of Nash-Williams, every planar graph can be covered by three trees. We show that every planar graph can be covered by two trees and a forest, and the maximum degree of the forest is at most 8. Stronger results are obtained for some special classes of planar graphs.
متن کاملOn the M-polynomial of planar chemical graphs
Let $G$ be a graph and let $m_{i,j}(G)$, $i,jge 1$, be the number of edges $uv$ of $G$ such that ${d_v(G), d_u(G)} = {i,j}$. The $M$-polynomial of $G$ is $M(G;x,y) = sum_{ile j} m_{i,j}(G)x^iy^j$. With $M(G;x,y)$ in hands, numerous degree-based topological indices of $G$ can be routinely computed. In this note a formula for the $M$-polynomial of planar (chemical) graphs which have only vertices...
متن کاملOn the Structure of Graphs with Large Minimum Bisection
Bounded degree trees and bounded degree planar graphs on n vertices are known to admit bisections of width O(logn) and O( √ n), respectively. We investigate the structure of graphs that meet this bound. In particular, we show that such a tree must have diameter O(n/ logn) and such a planar graph must have tree width Ω( √ n). To show the result for trees, we derive an inequality that relates the...
متن کاملComplexity of Finding Non-Planar Rectilinear Drawings of Graphs
Monotone Drawings of Graphs p. 13 Upward Geometric Graph Embeddings into Point Sets p. 25 On a Tree and a Path with No Geometric Simultaneous Embedding p. 38 Difference Map Readability for Dynamic Graphs p. 50 Maximizing the Total Resolution of Graphs p. 62 Plane Drawings of Queue and Deque Graphs p. 68 An Experimental Evaluation of Multilevel Layout Methods p. 80 Orthogonal Graph Drawing with ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM J. Discrete Math.
دوره 2 شماره
صفحات -
تاریخ انتشار 1989