arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1609.01221math.CO
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排除大theta图

Excluding a large theta graph

Guoli Ding, Emily Marshall

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27 pages, 8 figures
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摘要：

θ图，记为$θ_{a,b,c}$，是一个阶为$a+b+c-1$的图，由一对顶点和它们之间的三条独立路径组成，长度为$a$、$b$和$c$。我们提供了一个完整的表征图，不包含一个大的$θ_{a,b,c}$作为拓扑未成年人。更具体地说，我们描述了结构的$θ_{1,2,t}$ -，$θ_{2,2,t}$ -，$θ_{1,t,t}$ -，$θ_{2,t,t}$ -，和$θ_{t,t,t}$ -自由图，其中$t$是大的。主要结果是对大$t$的无$θ_{t,t,t}$图的一个刻画. $3$ -连通$θ_{t,t,t}$ -自由图是由$3$ -求和图构成的，这些图没有到某些平面图的长路. $2$ -连通的无$θ_{t,t,t}$图也可以用类似的方法通过2-和3-和来构造.这一结果意味着一个著名的定理罗伯逊和Chakravarti的图，没有一个键包含三个指定的边缘。

英文摘要：

A theta graph, denoted $θ_{a,b,c}$, is a graph of order $a+b+c-1$ consisting of a pair of vertices and three independent paths between them of lengths $a$, $b$, and $c$. We provide a complete characterization of graphs that do not contain a large $θ_{a,b,c}$ as a topological minor. More specifically, we describe the structure of $θ_{1,2,t}$-, $θ_{2,2,t}$-, $θ_{1,t,t}$-, $θ_{2,t,t}$-, and $θ_{t,t,t}$-free graphs where $t$ is large. The main result is a characterization of $θ_{t,t,t}$-free graphs for large $t$. The $3$-connected $θ_{t,t,t}$-free graphs are formed by $3$-summing graphs without a long path to certain planar graphs. The $2$-connected $θ_{t,t,t}$-free graphs are then built up in a similar fashion by 2- and 3-sums. This result implies a well-known theorem of Robertson and Chakravarti on graphs that do not have a bond containing three specified edges.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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