Let G = (V, E) be a simple, undirected graph. A set of vertices D is called an odd dominating set if |N [v] ∩D| ≡ 1(mod 2) for every vertex v ∈ V (G). The minimum cardinality of an odd dominating set is called the odd domination number of G, denoted by γ1(G). In this paper, several algorithmic and structural results are presented on this parameter for grids, complements of powers of cycles, and...

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...

A labeled graph is said to be weakly bipartite if the clutter of its odd cycles is ideal. Seymour conjectured that a labeled graph is weakly bipartite if and only if it does not contain a minor called an oddK5. A proof of this conjecture is given in this paper. Let G = (V;E) be a graph and E. We call edges in odd and edges in E even. The pair (G; ) is called a labeled graph. A subset L E(G) is ...

Let G = (V, E) be a finite, simple and undirected graph having v = |V (G)| and e = |E(G)|. A graph G with q edges is said to be odd-graceful if there is an injection f : V (G) → {0, 1, 2, . . . , 2q−1} such that, when each edge xy is assigned the label |f(x)−f(y)|, the resulting edge labels are {1, 3, 5, . . . , 2q−1}. Motivated by the work of Z. Gao [6], we have defined odd graceful labeling f...

Given a graph G = (V, E) and a set T ⊆ V , an orientation of G is called T -odd if precisely the vertices of T get odd in-degree. We give a good characterization for the existence of a T -odd orientation for which there exist k edge-disjoint spanning arborescences rooted at a prespecified set of k roots. Our result implies Nash-Williams’ theorem on covering the edges of a graph by k forests and...

A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)− f(y)| . A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V (G) to {0, 1, ..., 2n − 1} such that the induced weights are {1, 3, ..., 2n − 1}. We show here that any forest whose components are caterpillars is odd...

A diamond is a graph on 4 vertices with exactly one pair of non-adjacent vertices, and an odd hole is an induced cycle of odd length. If G,H are graphs, G is H-free if no induced subgraph of G is isomorphic to H. A clique-colouring of G is an assignment of colours to the vertices of G such that no inclusion-wise maximal clique of size at least 2 is monochromatic. We show that every (diamond, od...