图形功能

摘要：

设$G=(V,E)$是一个图，$A$是它的邻接矩阵.我们说顶点$y \in V$是顶点$x_1, \ldots, x_k \in V$的函数，如果存在$k$变量的布尔函数$f$，使得对于任何顶点$z \in V - \{y, x_1, \ldots, x_k\}$，$A(y,z)=f(A(x_1,z),\ldots,A(x_k,z))$。顶点$y$的功能性$fun(y)$是最小$k$，使得$y$是$k$顶点的函数。图$G$的函数$fun(G)$是$\max\limits_H\min\limits_{y\in V(H)}fun(y)$，其中在$G$的所有导出子图$H$上取最大值。在本文中，我们表明，功能同时推广其他几个图形参数，如退化或cnumer-width，通过证明有界退化或有界cnumer-width意味着有界功能。此外，我们表明，这种推广是适当的揭示类的图的无界退化和ck-width，其中功能是有界的常数。这包括置换图，单位区间图和线图。我们还观察到有界泛函蕴涵有界VC-维数，即有界VC-维数的图是有界泛函的图的推广，而且这种推广也是适当的。

英文摘要：

Let $G=(V,E)$ be a graph and $A$ its adjacency matrix. We say that a vertex $y \in V$ is a function of vertices $x_1, \ldots, x_k \in V$ if there exists a Boolean function $f$ of $k$ variables such that for any vertex $z \in V - \{y, x_1, \ldots, x_k\}$, $A(y,z)=f(A(x_1,z),\ldots,A(x_k,z))$. The functionality $fun(y)$ of vertex $y$ is the minimum $k$ such that $y$ is a function of $k$ vertices. The functionality $fun(G)$ of the graph $G$ is $\max\limits_H\min\limits_{y\in V(H)}fun(y)$, where the maximum is taken over all induced subgraphs $H$ of $G$. In the present paper, we show that functionality generalizes simultaneously several other graph parameters, such as degeneracy or clique-width, by proving that bounded degeneracy or bounded clique-width imply bounded functionality. Moreover, we show that this generalization is proper by revealing classes of graphs of unbounded degeneracy and clique-width, where functionality is bounded by a constant. This includes permutation graphs, unit interval graphs and line graphs. We also observe that bounded functionality implies bounded VC-dimension, i.e. graphs of bounded VC-dimension extend graphs of bounded functionality, and this extension is also proper.

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