Let F be a family of graphs. Two graphs G1 = (V1, E1), G2 = (V2, E2) are said to have the same F-structure if there is a bijection f : V1 → V2 such that a subset S induces a graph belonging to F in G1 if and only if its image f(S) induces a graph belonging to F in G2. We prove that graph H is perfect if and only if it has the {P3, P 3}-structure of some perfect graph G.

Kreweras’ conjecture [1] asserts that every perfect matching of the hypercube Qd can be extended to a Hamiltonian cycle. We [2] proved this conjecture but here we present a simplified proof. The matching graph M(G) of a graph G has a vertex set of all perfect matchings of G, with two vertices being adjacent whenever the union of the corresponding perfect matchings forms a Hamiltonian cycle. We ...

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a graph was introduced to look for obstruction sets beyond those indu...

For terms not defined here, the reader is referred to [6]. For a graph G=(V, E), we denote by G the complement of G. We use |(G) to denote the size of a largest clique in G and :(G) to denote the size of a largest stable set in G, or simply | and : when no confusion is possible. A graph G is said to be perfect if, for each induced subgraph H of G, H can be coloured with |(H) colours such that e...

The scarcity of fully-annotated data becomes the biggest obstacle that prevents many deep learning approaches from widely applied. Weakly-supervised visual which can utilize inexact annotations is developed rapidly to remedy such a situation. In this paper, we study weakly-supervised task achieving pixel-level semantic segmentation only with image-level labels as supervision. Different other me...

In this paper, we study the relationship between forbidden subgraphs and the existence of a matching. Let H be a set of connected graphs, each of which has three or more vertices. A graph G is said to beH-free if no graph inH is an induced subgraph of G. We completely characterize the setH such that every connected H-free graph of sufficiently large even order has a perfect matching in the foll...

Given two k-graphs H and F , a perfect F -packing in H is a collection of vertexdisjoint copies of F in H which together cover all the vertices in H. In the case when F is a single edge, a perfect F -packing is simply a perfect matching. For a given fixed F , it is often the case that the decision problem whether an n-vertex k-graph H contains a perfect F -packing is NP-complete. Indeed, if k ≥...

Let G = (V,E) be a connected undirected graph. For any vertex v ∈ V , the closed neighborhood of v is N [v] = {v} ∪ {u ∈ V | uv ∈ E }. For S ⊆ V , the closed neighborhood of S is N [S] = ⋃ v∈S N [v]. The subgraph weakly induced by S is 〈S〉w = (N [S], E ∩ (S × N [S])). A set S is a weakly-connected dominating set of G if S is dominating and 〈S〉w is connected. The weakly-connected domination numb...

Let α (G) denote the maximum size of an independent set of vertices and μ (G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is a perfect matching. If α (G) + μ (G) = |V (G)|, then G is called a König-Egerváry graph. A graph is unicyclic if it has a unique cycle. It is known that a maximum matching can be found in O(m •√n) time for a graph with n v...