Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.

翻译：中位图是链接的图表, 其中所有三个顶端都有一个独特的顶点, 属于这三个顶端中每对之间最短的路径。 在本文中, 我们提供一些对平面中位图的新的描述。 更具体地说, 当平面图$G$是一个以被禁止的子图和以美元表示的等离子图结构的中位图时, 我们确定一个平面图是一个中位图, 以美元表示禁止的子图和以美元表示的等离子图结构, 以及四个周期内外包含的美元子图, 与任意的G$嵌入图嵌入有关。 这些结果导致我们对平面平面图有新的定性, 以立方图和平面平面平面图表示的平面图。 这些图以立方块和平面平面平面图开始, 用立方平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。