## On infinite outerplanar graphs.(English)Zbl 0816.05025

Outplanar graphs are those with a planar embedding where all vertices lie on the boundary of the same face. The well-known necessary and sufficient condition for finite graphs to be outplanar is that they contain no subdivision of $$K_ 4$$ or $$K_{2,3}$$. This result also holds for infinite graphs. However, the situation is different if the infinite graph is prohibited from having vertices or edges accumulation; such graphs are called $$p$$-outplanar. The two graphs below, $$L_ 4$$ and $$L_{2,3}$$, are outerplanar, but not $$p$$-outerplanar. $\begin{matrix} \vdots\\ \bullet\\ \rlap{\strut\hskip3em $$L_ 4$$\strut}|\\ \bullet\\ |\\ \cdots\text{---}\bullet \text{---}\bullet \text{---}\bullet \text{--- }\bullet \text{---}\bullet \text{---}\bullet \text{---}\bullet \text{--- }\cdots \end{matrix}$
$\begin{matrix} \rlap{\strut\hskip3em $$L_{2,3}$$\strut}\bullet\\ \cdots\text{---}\bullet \text{---}\bullet \text{---}\bullet {\textstyle\diagup\diagdown\atop \textstyle\diagdown\diagup} \bullet\text{---}\bullet \text{---}\bullet \text{---}\cdots\\ \bullet \end{matrix}$ The main result of the paper is that a graph is $$p$$-outerplanar if and only if it has no subdivision of $$K_ 4$$, $$K_{2,3}$$, $$L_ 4$$, or $$L_{2,3}$$.
Reviewer: M.Marx (Pensacola)

### MSC:

 05C10 Planar graphs; geometric and topological aspects of graph theory 05C75 Structural characterization of families of graphs

### Keywords:

planar embedding; outplanar graphs
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