AI 中文总结
该研究针对部分立方体子类，改进了中位数图的警察数上界，建立了单纯复形图警察数的上下界，并确定了二分轮图等特殊单纯复形图的警察数精确值。
AI 中文摘要
警察与强盗游戏是图上的经典追捕-逃避游戏。对于图G，警察数c(G)是保证捕获G上的强盗所需的最少警察数量。尽管该参数已在若干基础图类中确定，但部分立方体及其子类的精确结果相对较少。我们首先根据树维数为每个有限中位数图M建立上界，这显著改进了Crawford和Iršič Chenoweth的界；该结果细化了此前用超立方体嵌入维数表示的上界，能给出小得多的估计值。随后研究部分立方体子类——单纯复形图的警察数：对于有限图G，单纯复形图S(G)以G的团（含空团）为顶点，当两个团恰好相差一个顶点时相邻。我们根据G的团数建立c(S(G))的一般下界，根据其色数建立一般上界。最后作为直接应用，确定了一些特殊单纯复形图——二分轮图、斐波那契立方体和卢卡斯立方体的警察数精确值。
英文摘要
The game of Cops and Robbers is a classical pursuit--evasion game on graphs. For a graph $G$, the cop number $c(G)$ is the minimum number of cops needed to guarantee the capture of a robber on $G$. Although this parameter has been determined for several fundamental graph classes, comparatively few exact results are known for partial cubes and their subclasses. We first establish an upper bound for every finite median graph $M$ in terms of its tree-dimension, which improves Crawford and Iršič Chenoweth's bound significantly. This result refines the previous upper bound expressed in terms of a hypercube embedding dimension and can give a substantially smaller estimate. Then we investigate the cop numbers of simplex graphs---a subclass of partial cubes. For a finite graph $G$, the simplex graph $S(G)$ has the cliques of $G$, including the empty clique, as its vertices, with two cliques adjacent whenever they differ in exactly one vertex. We establish a general lower bound for $c(S(G))$ in terms of the clique number of $G$ and a general upper bound in terms of its chromatic number. Finally, as direct applications, we determine the exact values of cop numbers of some special simplex graphs---bipartite wheels, Fibonacci and Lucas cubes.
Comments21 pages, 3 figures
