叶幂的盒性
Boxicity of Leaf Powers
摘要：
图G的盒度（box（G））定义为使得G是轴平行t维盒的交图的最小整数t。一个图G是k-叶幂，如果存在一棵树T，使得该树的叶子对应于图G的顶点，并且图G中的两个顶点相邻当且仅当它们在T中对应的叶子的距离至多为k。叶幂是强弦图的一个子类，在进化生物学中用于构建系统发育树。我们证明了对于k叶幂G，box（G）\leq k-1.我们还显示了这个约束的紧密性，通过构建一个k-叶功率等于k-1盒。这个结果意味着存在具有任意高盒度的强弦图，这有点违反直觉。
英文摘要：
The boxicity of a graph G, denoted as box(G) is defined as the minimum integer t such that G is an intersection graph of axis-parallel t-dimensional boxes. A graph G is a k-leaf power if there exists a tree T such that the leaves of the tree correspond to the vertices of G and two vertices in G are adjacent if and only if their corresponding leaves in T are at a distance of at most k. Leaf powers are a subclass of strongly chordal graphs and are used in the construction of phylogenetic trees in evolutionary biology. We show that for a k-leaf power G, box(G)\leq k-1. We also show the tightness of this bound by constructing a k-leaf power with boxicity equal to k-1. This result implies that there exists strongly chordal graphs with arbitrarily high boxicity which is somewhat counterintuitive.
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