带符号图的Lovász Theta参数与Theta体
A Lovász Theta Parameter and Theta Body for Signed Graphs
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中文总结 AI 辅助
该研究为带符号图的平衡着色提出Lovász型半定参数与凸角层级结构，证明平衡完美性弱于关联双覆盖图的完美性，相关构造与广义Theta数等概念相关。
中文摘要 AI 辅助
我们为带符号图的平衡着色引入了一种Lovász型半定参数。其同态目标是配备正交对合的单位球面：固定分量与反固定分量发挥不同作用，对一个端点应用对合即可实现切换。所得参数具有含两个半正定矩阵的对称形式及同样对称的对偶形式，同时它是双切换图负边构成的普通图的严格向量色数的一半。我们的第二个主要贡献是凸角的层级结构：从平衡诱导子图多面体出发，在原始顶点空间中定义稳定集、Theta及团不等式松弛的带符号类似物。带符号Theta体具有固有双矩阵描述，其全1规范即新的标量参数，普通稳定集层级可从带符号二角图精确恢复。对于全负符号，该构造成为最大诱导二分子图问题的松弛，且与广义Theta数相关。最后，我们提出平衡完美性概念，并证明其严格弱于关联双覆盖图的完美性。
英文摘要
We introduce a Lovász-type semidefinite parameter for balanced colouring of signed graphs. Its homomorphism target is a unit sphere equipped with an orthogonal involution: the fixed and anti-fixed components play different roles, while applying the involution to one endpoint realizes switching. The resulting parameter admits a symmetric formulation with two positive semidefinite matrices and an equally symmetric dual. It is also one half of the strict vector chromatic number of the ordinary graph formed by the negative edges of the double switching graph. Our second main contribution is a hierarchy of convex corners. Starting from the balanced induced subgraph polytope, we define signed analogues of the stable-set, theta, and clique-inequalities relaxations in the original vertex space. The signed theta body has an intrinsic two-matrix description, its all-ones gauge is the new scalar parameter, and the ordinary stable-set hierarchy is recovered exactly from signed digon graphs. For all-negative signatures, the construction becomes a relaxation of the maximum induced bipartite subgraph problem and is related to the generalized theta number. Finally, we propose a notion of balanced perfectness and show that it is strictly weaker than perfectness of the associated double-cover graph.
