We study the class of edge-coloured graphs arising from the graph-theoretic representation of quantum photonic experiments that generate multipartite W-states. Abstracting away physical amplitudes and phases, we introduce W-state graphs: matching-covered graphs equipped with a half-edge 2-colouring such that every perfect matching contains exactly one bichromatic edge and every vertex is incident with a red half-edge. Our main contribution is a complete structural characterization of W-state graphs. We show that a graph is a W-state graph if and only if each of its 3-connected components is a W-cone, a simple and rigid building block defined by a universal vertex and a factor-critical base. This characterization implies that no W-state graph is simple and yields a recognition algorithm running as fast as verifying whether a graph is matching-covered. We also show that the natural generalization to Dicke states encounters a complexity barrier: verifying one of the two Dicke state conditions is itself coNP-complete, resolving an open problem of Vardi and Zhang [IJCAI 2023]. Our results place W-state graphs firmly within classical matching theory and precisely delineate the combinatorial structures capable of realizing idealized W-states in the experiment-graph framework.

翻译：我们研究由量子光子实验中生成多体W态的图论表示所衍生的一类边染色图。通过抽象化物理振幅与相位，我们引入W态图：一类配备半边二染色且满足每个完美匹配恰好包含一条双色边、每个顶点关联一条红色半边的匹配覆盖图。主要贡献在于对W态图的结构刻画：图G是W态图当且仅当其每个三连通分支均为W锥体——一种由泛顶点与因子临界基构成的简单刚性构件。该刻画证明不存在简单W态图，并给出与验证匹配覆盖性同等高效的识别算法。进一步证明，向Dicke态的自然推广遭遇复杂性障碍：验证两个Dicke态条件之一即属于coNP-完全问题，解决了Vardi与Zhang [IJCAI 2023] 提出的公开问题。本研究将W态图严格纳入经典匹配理论框架，并精确刻画了实验图框架下能实现理想化W态的组合结构。