Removable Edges in Near-bricks

On removable edges in 3-connected cubic graphs

A removable edge in a 3−connected cubic graph G is an edge e = uv such that the cubic graph obtained from G \ {u, v} by adding an edge between the two neighbours of u distinct from v and an edge between the two neighbours of v disctinct from u is still 3−connected. Li and Wu [3] showed that a spanning tree in a 3−connected cubic graph avoids at least two removable edges, and Kang, Li and Wu [4]...

Contractible edges and removable edges in a graph with large minimum degree

On removable cycles through every edge

Mader and Jackson independently proved that every 2-connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G\E(C) is 2-connected. This paper considers the problem of determining when every edge of a 2-connected graph G, simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of tw...

Brick Generation and Conformal Subgraphs

A nontrivial connected graph is matching covered if each of its edges lies in a perfect matching. Two types of decompositions of matching covered graphs, namely ear decompositions and tight cut decompositions, have played key roles in the theory of these graphs. Any tight cut decomposition of a matching covered graph results in an essentially unique list of special matching covered graphs, call...

Removable Edges in Longest Cycles of 4-Connected Graphs

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G e; second, for all vertices x of degree 3 in G e, delete x from G e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by G e. If G e is 4-conn...