A graph G is called g-perfect if, for any induced subgraph H of G, the game chromatic number of H equals the clique number of H. A graph G is called g-col-perfect if, for any induced subgraph H of G, the game coloring number of H equals the clique number of H. In this paper we characterize the classes of g-perfect resp. g-col-perfect graphs by a set of forbidden induced subgraphs. Moreover, we ...

We call an edge e of a perfect graph G critical if G e is imperfect and call e anticritical if G + e is imperfect. The present paper surveys several questions in this context. We ask in which perfect graphs critical and anticritical edges occur and how to detect such edges. The main result by Hougardy & Wagler [32] shows that a graph does not admit any critical edge if and only if it is Meyniel...

Let G be a cubic graph with a perfect matching. An edge e of G is a forcing edge if it is contained in a unique perfect matching M , and the perfect matching M is called uniquely forced. In this paper, we show that a 3-connected cubic graphwith a uniquely forced perfect matching is generated from K4 via Y → 1-operations, i.e., replacing a vertex by a triangle, and further a cubic graph with a u...

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 tiroes the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduc...

A graph is called t-perfect if its stable set polytope is defined by nonnegativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect.

The Hungarian method is an e cient algorithm for nding a minimal cost perfect matching in a weighted bipartite graph. This paper describes an e cient algorithm for nding all minimal cost perfect matchings. The computational time required to generate each additional perfect matching is O(n(n + m)); and it requires O(n+m) memory storage. This problem can be solved by algorithms for nding the Kth-...

‎An  oriented perfect path double cover (OPPDC) of a‎ ‎graph $G$ is a collection of directed paths in the symmetric‎ ‎orientation $G_s$ of‎ ‎$G$ such that‎ ‎each arc‎ ‎of $G_s$ lies in exactly one of the paths and each‎ ‎vertex of $G$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete‎ ‎Math‎. ‎276 (2004) 287-294) conjectured that ...