斯特布尔-德明格图：特征化
Sterboul-Deming Graphs: Characterizations
AI总结：
本文研究了斯特布尔-德明格图的特征化，通过构造性算法和加拉伊-埃德蒙兹分解，揭示了其结构特性及广泛的应用范围。
摘要：
一个图被称为斯特布尔-德明格图，如果KE(G)=∅，即每个顶点都属于一个posy或一个flower（由斯特布尔、德明和埃德蒙兹引入的结构）。这些图可以看作是 König-Egerváry 图的结构对应物。在本文中，我们提出了几种斯特布尔-德明格图的特征化方法。我们首先研究了具有完美匹配且具有唯一完美匹配的图的情况，提供了一种构造性算法来获得分解(SD(G), KE(G))。然后，我们通过加拉伊-埃德蒙兹分解扩展分析到一般情况。此外，我们还证明斯特布尔-德明格图的类别非常广泛：它包含所有具有{C_n : n为奇数}-因子的图，为识别此类图提供了一个简单的结构准则。这些结果建立了经典分解定理与非König-Egerváry图内部结构之间的新联系。
英文摘要：
A graph is said to be a Sterboul--Deming graph if $KE(G)=\emptyset$, that is, if every vertex of $G$ belongs to a posy or a flower (structures introduced by Sterboul, Deming, and Edmonds). These graphs can be regarded as the structural counterparts of König--Egerváry graphs. In this paper, we present several characterizations of Sterboul--Deming graphs. We first study the case of graphs with a perfect matching and with a unique perfect matching, providing a constructive algorithm to obtain the decomposition $(SD(G), KE(G))$. Then, we extend the analysis to the general case through the Gallai--Edmonds decomposition. In addition, we show that the class of Sterboul--Deming graphs is remarkably broad: it contains all graphs having a $\{C_n : n \textnormal{ odd}\}$-factor, providing a simple structural criterion for identifying such graphs. These results establish new connections between classical decomposition theorems and the internal structure of non--König--Egerváry graphs.
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