图的$4\times 4$矩阵划分的计数
Counting $4\times 4$ Matrix Partitions of Graphs
摘要：
给定对称矩阵$M\in \{0,1,*\}^{D\times D}$，图$G$的$M$ -划分是从$V(G)$到$D$的函数，使得$G$的任何边都不映射到$M$的$0$，并且没有任何非边映射到$1$ .我们给出了一个计算机辅助的证明，当$|D|=4$，计数的$M$ -分区的输入图的问题是在FP或#P-完全。可追溯性证明减少计数列表$M$分区的相关问题，棘手的是使用一个小工具建设和插值。我们使用一个计算机程序来确定这两种情况下，除了少数矩阵，我们手动解决建立二分法。我们猜想，二分法也适用于$|D|>4$。更具体地说，我们推测，对于任何对称矩阵$M\in\{0,1,*\}^{D\times D}$，计数$M$ -分区的复杂性是相同的计数列表$M$ -分区的相关问题。
英文摘要：
Given a symmetric matrix $M\in \{0,1,*\}^{D\times D}$, an $M$-partition of a graph $G$ is a function from $V(G)$ to $D$ such that no edge of $G$ is mapped to a $0$ of $M$ and no non-edge to a $1$. We give a computer-assisted proof that, when $|D|=4$, the problem of counting the $M$-partitions of an input graph is either in FP or is #P-complete. Tractability is proved by reduction to the related problem of counting list $M$-partitions; intractability is shown using a gadget construction and interpolation. We use a computer program to determine which of the two cases holds for all but a small number of matrices, which we resolve manually to establish the dichotomy. We conjecture that the dichotomy also holds for $|D|>4$. More specifically, we conjecture that, for any symmetric matrix $M\in\{0,1,*\}^{D\times D}$, the complexity of counting $M$-partitions is the same as the related problem of counting list $M$-partitions.
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