in this paper we define the removable cycle that, if $im$ is a class of graphs, $gin im$, the cycle $c$ in $g$ is called removable if $g-e(c)in im$. the removable cycles in eulerian graphs have been studied. we characterize eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for eulerian graph to have removable cycles h...

In this paper we define the removable cycle that, if $Im$ is a class of graphs, $Gin Im$, the cycle $C$ in $G$ is called removable if $G-E(C)in Im$. The removable cycles in Eulerian graphs have been studied. We characterize Eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for Eulerian graph to have removable cycles h...

‎in this paper we defined the vertex removable cycle in respect of the following‎, ‎if $f$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$g in f $‎, ‎the cycle $c$ in $g$ is called vertex removable if $g-v(c)in in f $.‎ ‎the vertex removable cycles of eulerian graphs are studied‎. ‎we also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

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...

The lower bound on the length of 'removable' cycles in Theorem 1 is essentially best possible since there exist 2-connected simple graphs of minimum degree k whose longest 'removable' cycle has length k + i. Moreover, for the special case k = 4, we can construct 2-connected, 4-regular simple graphs whose longest 'removable' cycle has length four. The following counterexample which was independe...

Kriesell proved that every almost critical graph of connectivity 2 nonisomorphic to a cycle has at least 2 removable ears of length greater than 2. We improve this lower bound on the number of removable ears. A necessary condition for critically 2-connected graphs in terms of a forbidden minor is obtained. Further, we investigate properties of a special class of critically 2-connected series-pa...

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...