# On Orthogonal Double Covers and Decompositions of Complete Bipartite Graphs by Caterpillar Graphs
## 1 Introduction  
Graph theory occupies a special place in mathematics and is used in many areas of engineering to solve various problems, from describing the topologies of communication networks of autonomous vehicles [1] to medical research in eating disorders [2] and social interaction during pandemics [3]. Caterpillar graphs [4] are emphasized in graph theory due to their properties. They have gained their popularity since the 1970s and are still being studied. Moreover, they are studied not only in the context of their mathematical properties [5; 6; 7; 8; 9], but are also especially popular in chemistry [10], where they are used to describe benzenoid hydrocarbon molecules under the names benzenoid trees [11; 12], alkane graphs, and Gutman trees [10; 13]. Caterpillar graphs are also used to describe more complex graphs (for example, corona graphs [14]), leaf realization problems on graphs [15], and minimum spanning tree problems on graphs [16]. Currently, the use of such graphs in coding theory [17; 18; 19; 20] is very popular, especially in the development of technologies for post-quantum cryptography [21]. The designations used in the paper are shown in Table 1.  
Consider that \(H\cong K_{x,x}\) and \(\mathcal{G}=\{T_{0},T_{1},\ldots,T_{x-1},A_{0},A_{1},\ldots,A_{x-1}\}\) be a collection that describes the \(2x\) isomorphic subgraphs, then we can call \(\mathcal{G}\) an orthogonal double cover \((ODC)\) of \(K_{x,x}\) by \(G\), if the following two conditions are satisfied: (i) every edge of \(K_{x,x}\) exists in two isomorphic subgraphs of \(\mathcal{G}\) and (ii) for \(\alpha,\beta\in\{0,1,\ldots,x-1\}\), \(\left|E(T_{a})\cap E\big{(}A_{\beta}\big{)}\right|=1\), and for \(\alpha\
eq\beta\), \(\left|E(T_{a})\cap E\big{(}T_{\beta}\big{)}\right|=\left|E(A_{\alpha})\cap E \big{(}A_{\beta}\big{)}\right|=0\). The significance of graph decompositions can be exemplified as follows. By recognizing a graph's structure through dividing a graph into more easily understood parts, decompositions can aid in our understanding of its structure [22]. By breaking down a graph into its connected parts, the number of distinct subgraphs present in the network can be defined. Via decompositions, the degree distribution and clustering coefficient of a graph can be identified. By breaking a graph down into its degree sequence, the distribution of degrees among the graph's vertices can also be determined. Decompositions can be applied to algorithms for the network flow analysis and graph coloring [23]. It is possible to find the maximum flow in a graph by breaking it down into its edge-disjoint path ways. Decompositions are a topic of ongoing research in a graph theory causing numerous unanswered questions. Decompositions can be studied to gain new knowledge and understanding in this field. Furthermore, decompositions of graphs are a useful tool for understanding and analyzing graphs in general, and they have many real-world uses in many disciplines, including computer science [24], mathematics [25; 26], and social science. The existence of \(OD\)Cs has been considered and generalized by many authors [27; 28; 29]. The importance of \(ODC\) problems stems from solving database optimization problems; for example, when we look for specific information in a very large database, it takes too much time to find that information [30]. In [24], \(OD\)Cs are related to several problems such as the statistical design of experiments and design theory. It can also be used to solve the routing problem in various network systems whose topologies are large regular graphs; for example, see works [31; 32]. The specialists have solved the \(ODC\) problem for various graphs such as circulant graphs, complete graphs, and complete bipartite graphs [33]. Note that these graphs are usually very large. Definition 1: Let \(\geq 2\) be integer. A caterpillar graph \(C_{l}(w_{1},w_{2},\ldots,w_{l})\) was obtained from a path \(P_{l}=u_{1}u_{2}\ldots u_{l}\) by appending \(w_{i}\geq 0\) necklace vertices to each \(u_{i}\), \(1\leq i\leq l\). **Example 1**.