分裂图的矩阵划分

Minimal obstructions for a matrix partition problem in chordal graphs

If $M$ is an $m \times m$ matrix over $\{ 0, 1, \ast \}$, an $M$-partition of a graph $G$ is a partition $(V_1, \dots V_m)$ such that $V_i$ is completely adjacent (non-adjacent) to $V_j$ if $M_{ij} = 1$ ($M_{ij} = 0$), and there are no further restrictions between $V_i$ and $V_j$ if $M_{ij} = \ast$. Having an $M$-partition is a hereditary property, thus it can be characterized by a set of minimal obstructions (forbidden induced subgraphs minimal with the property of not having an $M$-partition). It is known that for every $3 \times 3$ matrix $M$ over $\{ 0, 1, \ast \}$, there are finitely many…

关于分裂图边着色与全着色的新结果

New Results on Edge-coloring and Total-coloring of Split Graphs

本文针对 σ=2 的分裂图，利用按伸缩指数划分的结构，刻画 Class 2 与 Type 2 图，并为 Class 1、Type 1 图提供边着色和全着色的多项式算法。

A combinatorial algorithm for computing the entire sequence of the maximum degree of minors of a generic partitioned polynomial matrix with $2 \times 2$ submatrices

In this paper, we consider the problem of computing the entire sequence of the maximum degree of minors of a block-structured symbolic matrix (a generic partitioned polynomial matrix) $A = (A_{αβ} x_{αβ} t^{d_{αβ}})$, where $A_{αβ}$ is a $2 \times 2$ matrix over a field $\mathbf{F}$, $x_{αβ}$ is an indeterminate, and $d_{αβ}$ is an integer for $α= 1,2,\dots, μ$ and $β= 1,2,\dots,ν$, and $t$ is an additional indeterminate. This problem can be viewed as an algebraic generalization of the maximum weight bipartite matching problem. The main result of this paper is a combinatorial $O(μν\min\{μ, ν\}…

Exploiting Equitable Partitions for Efficient Block Triangularization

In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block by its row sum yields a smaller matrix whose multiset of eigenvalues is contained in the initial spectrum. We generalize this approach to weighted row sums and rectangular matrices and derive an efficien…

单极图和广义分裂图的算法

Algorithms for Unipolar and Generalized Split Graphs

一个图$G=(V,E)$是{ \it单极图}，如果存在一个划分$V=V_1 \cup V_2$，使得$V_1$是一个团，并且$V_2$诱导团的不交并. { \it广义分裂图}的补闭类是那些图$G$，使得$G$ { \it}或}的补是单极的。广义分裂图是完美图的一个大子类。事实上，几乎所有的无$C_5$图（因此，几乎所有的完美图）都是广义分裂图。在本文中，我们提出了一个单极图的识别算法，利用给定的图的最小三角剖分，并产生一个分区时，存在。我们的算法的运行时间为O（$nm^\prime$），其中$m^\prime$是给定图的最小三角剖分中的边数.该算法可以在O（$nm' + n\OL{m}'$）= O（$n^3$）时间内识别出广义分裂图.当给定单极划分时，我们给出了在O（$n+m$）时间内求最大独立集和最小团覆盖以及在O（$n^{2.5}/\log n$）时间内求最大团和最小真染色的算法.这些算法产生算法的四个优化问题的广义分裂图，具有相同的最坏情况的时间范围。我们还证明了单极图的完美码问题是NP-完全的。