Abstract: Undirected graph is a elementary concept of graph theory. Graph semigroup is a concept proposed in 1991. A concept of undirected graph language was proposed with undirected graph and graph semigroup. Furthermore, a subclass of undirected graph language, plane graph language was studied. It is proved that a graph language is a plane graph language if and only if it contains no subdivision graph language of K5 language or K3,3 language. In addition,four open problems were proposed. One of them:what is the relations between undirected graph language and regular language,context free language, context sensitive language, phrase structure language, respectively.

Key words: Undirected graph, Graph semigroup, Undirected graph language, Plane graph language