A (Euclidean) greedy drawing of a graph is a drawing in which, for any two vertices $s,t$ ($s \neq t$), there is a neighbor vertex of $s$ that is closer to $t$ than to $s$ in the Euclidean distance. Greedy drawings are important in the context of message routing in networks, and graph classes that admit greedy drawings have been actively studied. N\"{o}llenburg and Prutkin (Discrete Comput. Geom., 58(3), pp.543-579, 2017) gave a characterization of greedy-drawable trees in terms of an inequality system that contains a non-linear equation. Using the characterization, they gave a linear-time recognition algorithm for greedy-drawable trees of maximum degree $\leq 4$. However, a combinatorial characterization of greedy-drawable trees of maximum degree 5 was left open. In this paper, we give a combinatorial characterization of greedy-drawable trees of maximum degree $5$, which leads to a complete combinatorial characterization of greedy-drawable trees. Furthermore, we give a characterization of greedy-drawable pseudo-trees.

翻译：图的一幅（欧几里得）贪心绘制是指这样一种绘制：对于任意两个顶点$s,t$（$s \neq t$），存在$s$的某个邻居顶点，该顶点到$t$的欧几里得距离小于其到$s$的距离。贪心绘制在网络消息路由中具有重要意义，可支持贪心绘制的图类一直受到积极研究。Nöllenburg与Prutkin（Discrete Comput. Geom., 58(3), pp.543-579, 2017）通过包含一个非线性方程的不等式系统给出了可贪心绘制的树的刻画。利用这一刻画，他们提出了最大度$\leq 4$的可贪心绘制树的线性时间识别算法。然而，最大度为5的可贪心绘制树的组合刻画仍未解决。本文给出了最大度为$5$的可贪心绘制树的组合刻画，从而完成了可贪心绘制树的完全组合刻画。此外，我们还给出了可贪心绘制的伪树的刻画。