符号Colin de Verdière参数$ν$为2的符号图的图子式刻画
A graph minors characterization of signed graphs whose signed Colin de Verdière parameter $ν$ is two
摘要：
有符号图是一对$(G,Σ)$，其中$G=(V,E)$是具有$V=\{1,...,n\}$和$Σ\subseteq E$的图（其中允许平行边，但不允许循环）。$Σ$中的边称为奇数，其他边称为偶数。通过$S(G,Σ)$，我们表示所有对称$n\times n$矩阵的集合$A=[a_{i,j}]$，如果$i$和$j$仅由偶数边连接，则$a_{i,j}<0$，如果$i$和$j$仅由奇数边连接，则$a_{i,j}>0$，如果$i$和$j$由偶数边和奇数边连接，则$a_{i,j}\in \mathbb{R}$，如果$i\not=j$和$i$和$j$不相邻，则$a_{i,j}=0$，并且$a_{i,i} \in \mathbb{R}$用于所有顶点$i$。符号图$(G,Σ)$的参数$ν(G,Σ)$是具有强Arnold性质的半正定矩阵$A\in S(G,Σ)$的最大零值。用$K_3^=$表示由$(K_3,\emptyset)$通过向每个偶数边并行添加一个奇数边而获得的带符号图。本文证明了符号图$(G,Σ)$有$ν(G,Σ)\leq 2$当且仅当$(G,Σ)$没有子同构于$(K_4,E(K_4))$或$K_3^=$ .
英文摘要：
A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,...,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for all vertices $i$. The parameter $ν(G,Σ)$ of a signed graph $(G,Σ)$ is the largest nullity of any positive semidefinite matrix $A\in S(G,Σ)$ that has the Strong Arnold Property. By $K_3^=$ we denote the signed graph obtained from $(K_3,\emptyset)$ by adding to each even edge an odd edge in parallel. In this paper, we prove that a signed graph $(G,Σ)$ has $ν(G,Σ)\leq 2$ if and only if $(G,Σ)$ has no minor isomorphic to $(K_4,E(K_4))$ or $K_3^=$.
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