A Study on Co – odd (even) Sum Degree Edge Domination Number in Graphs

: An edge dominating set 𝑇  𝐸(𝐺) of a graph 𝐺 = (𝑉, 𝐸) is said to be an odd (even) sum degree edge dominating set (osded (esded) - set) of G if the sum of degree of all the edges in T is an odd (even) number. The odd (even) sum degree edge domination number 𝛾 𝑜𝑠𝑑′ (𝐺) (𝛾 𝑒𝑠𝑑′ (𝐺)) is the minimum cardinality taken over all odd (even) sum degree edge dominating sets of G and is defined as zero if no such odd (even) sum degree edge dominating set exists in G. In this paper, the odd (even) sum degree domination concept is extended on the co-dominating set E-T of a graph G, where T is an edge dominating set of G.


Introduction:
Let G = (V, E) be a simple, connected, finite, and undirected graph. The maximum and minimum degree of a graph G is respectively denoted by ∆(G) and δ(G). The cardinality of the vertex (edge) set of a graph G is called the order (size) of G and is denoted by p(q). A graph with p vertices and q edges is called a (p, q) graph. A regular graph is a graph where each vertex has the same number of neighbors. If 't' is a vertex of G, then its degree is denoted by deg(t). A set T  E(G) is an edge dominating set if every edge in E(G) -T is adjacent to at least one edge in T. The edge domination number '(G) is the minimum cardinality of an edge dominating set in G. The edge dominating set with cardinality '(G) is denoted as ' -set of G. The edge set E (G) -T is said to be a co-edge dominating set 1-3 of G. All the basic graph terminologies are used in the sense of Harary 4 . In the year 2003, the odd domination number of graph G is introduced by Yair Caro and William F. Klostermeyer 5 . The odd geo-domination number of a graph is introduced by Anto Kinsley A and Karthika K John in the year 2020 6 . Motivated by the notion of the above parameters and their applicability, the odd (even) sum degree edge dominating set (oded (eded) -set) is introduced by posting odd (even) condition on the sum of degree of edges of edge dominating set of a graph G. In this paper, by extending the above concept on coedge dominating set, the co-odd (even) sum degree edge dominating set is defined for a graph G 7-9 . Then the corresponding co-odd (even) sum degree edge domination number and value are defined and studied. All the graphs considered in this article are referred from Joshep A. Gallian 10 . In this paper, coodd (even) sum degree edge domination numbers are found for some standard classes of graphs such as Path, Cycle, Wheel, Comb, Star, Crown, Friendship, Helm, Triangular Snack, Fan, Book, Dumbbell, Flag, Todpole 11 and Caterpillar graphs. Further, the bounds of the above parameters are obtained and the relationships between some of the existing edge domination parameters are studied.
Every graph has at least one cesdedset since every graph has an even number of odd degree edges.

2.
Cosded-set need not exist in all graphs, for example, Cycle C n , n ≥ 3 does not have a cosdedset.

Proof:
Let G be a path graph with at least three vertices. Let E(G) = {t 1 , t 2 , t 3 , … ,t n-1 }be the edge set of G with deg(t 1 ) = deg(t n-1 ) = 1 and the remaining edges have degree 2. Claim (i): Let X 1 = {t 3i-1 / i =1, 2, … ,⌊ −1 3 ⌋} and X 2 = {t n-1 } be two edge sets of G. Note that X 1 is an edge dominating set of G and X 1 U X 2 is an odd degree edge dominating set of G. Hence, On the other hand, suppose Y is a ′ cosd set of G. Then for edge domination Y must have at least 3 ⌋} edges and for the odd sum degree edge domination Y must contain {t n-1 }. It completes Claim (i). Claim (ii): When n = 3, the edge set X = {t 1 , t 2 } is a ′ cesdset of G and hence, ′ cesd (G) = 2. When n = 4, the edge set X = {t 2 } itself a ′ cesdset of G and hence, ′ cesd (G) =1.
On the other hand, suppose Y is a ′ cesd set of G. Then Y must contain at Then the result in Claim (ii) is proved from Eq. 3 and 4. Further, if T is ′ cesdset of G then the co-even sum degree edge domination value It completes Claim (ii). □ Theorem. 2: For the cycle graph C n, n ≥ 3

Proof:
Let G be a cycle graph C n with at least three vertices. Let E(G) = {t 1 ,t 2 ,…,t n }be the edge set of G. Since all the edges in E(G) are in even degrees gives ′ cosd (C n )= 0. Let T = {t 3i-2 / i=1,2,..., ⌊ +2 3 ⌋} be an edge set of G. Note that T is an edge dominating set of G and also T is an even degree edge dominating set of G.

Proof:
Let G be a Pan graph with at least four vertices. It completes Claim (ii).

Bounds and Characterization of Odd (Even) Sum Degree Edge Dominating Sets of G The following result gives the relation between the edge domination number and co-odd (even) degree edge domination number.
Theorem. 4: For any graph G, Since every cosded-set and cesded-set is an edge dominating set of G, proves the results immediately. For (a), the bound is sharp for star S 3 .

Proof:
Since G has only even degree edges gives there is no cosdedset in G. Hence, ′ cosd (G) = 0. At the same time, every ′set of G is an ′ cesd -set, and hence ′ cesd (G) = (G). not cesdedset. Therefore, T e has an even number of odd degree edges. Since the sum of even numbers is even given T e may contain any number of even degree edges. Further, the above two combinations also give T-T e a cesdedset of G. It provestheclaim. □ Theorem. 7: Let G be a graph with distinct minimum and maximum edge degrees '(G) and ∆'(G). If T o and T e are ′ cosdset and ′ cesdset of G then (i) Since δ′(G) and ∆′(G) are the minimum and maximum edge degrees of graph G, gives δ′(G) ≤ deg (t) ≤ ∆′(G), if δ′ ≠ ∆′ and e ∈G. Therefore, Hence, from Eq.2 and 3, By similar arguments, one can prove the result, For (i), the lower and upper bound is sharp for star S 3 . For (ii), the lower and upper bound is sharp for cycle C 3 . □ Theorem. 8: Let G be a graph. Then (i) G is a cosdedgraph if and only if G has at least one odd degree edge. (ii) G is a cesdedgraph if and only if G has at least even numbers of odd degree edges (or) degree of all the edges are even. Proof: Claim (i): Let G is a cosdedgraph, then there exists a cosdedset T-T o of G. Assume that G has no odd degree edge. Since G is connected with at least two edges gives G has only edges of even degree. Then there is no cosdedset existing in G, which contradicts that G has a cosded-set T-T o . Therefore, G has at least one odd degree edge. The converse is obvious. Claim (ii): Let G is a cesdedgraph, then there exists a cesdedset T-T e of G. Suppose G has odd degree edges. Then for the existence of T-T e , G must have an even number of odd degree edges. On the other hand, suppose G has only even degree edges then the result is immediate. The converse of the result is obvious. □ Theorem. 9: Let G is a graph with an osdedset T o and esdedset T e . If ' (t) be an even number then (a) T -T o is a cosdedset of G and (b) T -T e is a cesdedset of G Proof: Let S' cosd (t) = 2m, m∈N. Then S' osd (t) + S' cosd (t) = 2m. Then S' osd (t) = 2m -S' cosd (t). Since T o be an osdedset of G given ∑ deg(e) is also an odd number. This shows that T -T o be a cosded set of G. By a similar argument, one can prove that T -T e is a cesdedset of G. □

Conclusion:
In this paper, the exact values of the co-odd (even) sum degree edge domination number and coodd (even) sum degree edge domination value are found for some standard classes of graphs described below: The bounds of the co-odd (even) sum degree edge domination number are obtained. The co-odd (even) sum degree edge dominating sets are characterized. The relationships with other edge domination parameters are also determined.