Some Characterizations and NP-Complete Problems for Power Cordial Graphs

. A power cordial labeling of a graph G � ( V ( G ) , E ( G )) is a bijection f : V ( G ) ⟶ 1 , 2 , . .. , | V ( G )| { } such that an edge e � uv is assigned the label 1 if f ( u ) � ( f ( v )) n or f ( v ) � ( f ( u )) n , for some n ∈ N ∪ 0 { } and the label 0 otherwise, and satisfy the number of edges labeled with 0 and the number of edges labeled with 1 difer by at most 1. Te graph that admits power cordial labeling is called a power cordial graph. In this paper, we derive some characterizations of power cordial graphs as well as explore NP-complete problems for power cordial labeling. Tis work also rules out any possibility of forbidden subgraph characterization for power cordial labeling.


Introduction
We begin with a simple, fnite, and undirected graph G � (V(G), E(G)). Te order, size, and girth of a graph are the number of vertices, the number of edges, and the length of the shortest cycle contained in the graph, respectively. A graph is triangle-free if no three vertices form a cycle. A graph is k-cycle-free if no k vertices form a cycle. For all standard terminologies, notations, and terms not defned here, we refer to Clark and Holton [1]. We will give a brief summary of defnitions which are useful for the present investigations.

Defnition 1.
A graph labeling is an assignment of integers to the vertices or edges or both subject to certain condition(s). If the domain of the mapping is the set of vertices (edges), then the labeling is called a vertex labeling (an edge labeling).
For the bibliographic references as well as an extensive survey of various graph labelings, we refer to Gallian [2].
Labeled graphs have applications in many diferent felds such as coding theory, communication networks, and determination of optimal circuit layouts. A study on the wide variety of applications of diferent graph labelings have been explored by Yegnanarayanan and Vaidhyanathan [3].
In 2022, Barasara and Takkar [4] have introduced power cordial labeling of graphs and defned it as follows: Defnition 2. For a graph G � (V(G), E(G)), the power cordial labeling is defned as a bijection f: { } is given by the following: which satisfy the condition |e f (0) − e f (1)| ≤ 1, where e f (0) and e f (1) are the number of edges labeled with 0 and 1, respectively. Te graph that admits a power cordial labeling is called a power cordial graph.
In the same paper, we have investigated power cordial labeling for path, cycle, complete graph, wheel, tadpole, complete bipartite graph K 2,n , complete bipartite graph K 3,n , star, and bistar. In [5], we have discussed power cordial labeling for helm, fower, gear, fan, and jewel graphs as well as larger graphs obtained from the star and bistar using graph operations. In [6], we have derived results related to power cordial labeling of graphs in the context of vertex switching operation.
Tere are three types of problems that can be considered in this area, which are as follows: (1) How power cordiality is afected under various graph operations (2) Constructing new families of power cordial graphs by fnding suitable labeling (3) Given a graph theoretic property P, characterizing the class of graphs with property P that are power cordial As not all graphs are power cordial graphs, it is interesting to study which graph theoretic property afects or is afected by power cordial labeling.
In this paper, we focus on the problem of the third type. In Section 2, we give some characterizations of power cordial graphs. While in Section 3, we study NP-complete problems in the context of power cordial labeling.

Some Characterizations of Power Cordial Graphs
In this section, we discuss embedding in the context of power cordial labeling and derive some characterizations of power cordial graphs.
Theorem 3. Any graph G can be embedded as an induced subgraph of a power cordial graph.
Proof. Let G be the graph with p vertices and q edges. Assign labels 1, 2, . . . , p to the vertices of G. Let E i be the set of edges with label i and e f (0) − e f (1) � r.
We consider following three cases: Terefore, in Case 2 and Case 3, the constructed supergraph H satisfes the conditions for the power cordial graph.
Hence, any graph G can be embedded as an induced subgraph of a power cordial graph. □ Corollary 4. Any tree T can be embedded as an induced subgraph of a power cordial tree.
Proof. If T is a tree, then the supergraph T ′ constructed in Teorem 3 is a power cordial tree. Hence, the result. □

Corollary 5. Any acyclic (unicyclic) graph G can be embedded as an induced subgraph of an acyclic (a unicyclic) power cordial graph.
Proof. If G is an acyclic graph (a unicyclic), then the supergraph H constructed in Teorem 3 is an acyclic (a unicyclic) power cordial graph. Hence, the result. □ Corollary 6. Any triangle-free (k-cycle free) graph G can be embedded as an induced subgraph of a triangle-free (a k-cycle free) power cordial graph.
Proof. If G is a triangle-free (a k-cycle free) graph, then the supergraph H constructed in Teorem 3 is a triangle-free (a k-cycle free) power cordial graph. Hence, the result. □ Corollary 7. Any graph G having girth n can be embedded as an induced subgraph of a power cordial graph H having girth n.
Proof. If G is a graph having girth n, then the supergraph H constructed in Teorem 3 is a power cordial graph having girth n. Hence, the result.

Corollary 8. Any planar (connected) graph G can be embedded as an induced subgraph of a planar (a connected) power cordial graph.
Proof. If G is a planar (a connected) graph, then the supergraph H constructed in Teorem 3 is a planar (a connected) graph. Hence, the result. □ Theorem 9. Given a positive integer n, there is a power cordial graph G, which has n vertices.
Hence, given a positive integer n, there is a power cordial graph G, which has n vertices. □ For n � 10, a graph G with 10 vertices and its power cordial labeling is shown in Figure 1. Hence, if G is a power cordial graph of even size, then G − e is also a power cordial graph for all e ∈ E(G). □ Theorem 11. If G is a power cordial graph of odd size, then G − e is also a power cordial graph for some e ∈ E(G).
Proof. Let 2q + 1 be the size of the power cordial graph G. Hence, for G be a power cordial graph of even size and G ≇ K n , then adding an edge between any two nonadjacent vertices of G is also a power cordial graph. □ Theorem 13. Let G be any power cordial graph of order m and K 2,n be a bipartite graph with the bipartition V(K 2,n ) � V 1 ∪ V 2 with V 1 � x 1 , x 2 and V 2 � y 1 , y 2 , . . . , y n . Te graph G * K 2,n obtained by identifying x 1 of K 2,n with the vertex having label 1 in G and x 2 of K 2,n with the vertex having label as the largest prime number p such that p ≤ m in G. Ten, G * K 2,n is a power cordial graph.
Proof. Let G be any power cordial graph of order m. Let v ′ and v ″ be the vertices having label 1 and the largest prime number p ≤ m, respectively, in graph G.
Ten, identify the vertices x 1 and x 2 of K 2,n with v ′ and v ″ , respectively. Let the graph so obtained be G * K 2,n .
Case 1: for m + n < p 2 . Te labels of y i , for each i, does not contain power of p. Tus, all the edges of K 2,n incident with v ′ will receive label 1 and all the edges of K 2,n incident with v ″ will receive label 0. So, the edges of K 2,n contribute equally to e f (0) and e f (1) in G * K 2,n . Tus, G * K 2,n is a power cordial graph. Case 2: for m + n ≥ p 2 . Let q be the largest prime number such that q ≤ m + n. Suppose y k has label q. Ten, interchange the labels of Journal of Mathematics v ″ and y k . Ten, according to Case 1, G * K 2,n is a power cordial graph.
Hence, G * K 2,n is a power cordial graph.

□
Cycle C 4 and the graph C 4 * K 2,4 and their power cordial labeling are shown in Figure 2.
Proof. Let G be the graph of size n and v be a vertex of G such that ⌊n/2⌋ ≤ d(v) ≤ ⌈n/2⌉.
We consider two cases as follows: Case 1: when n is even. Assign label 1 to vertex v and label the remaining vertices in such a way that they do not generate edge label 1 (except for the edges having one of the end vertex v). Ten, e f (0) � n/2 and e f (1) � n/2. Tus, f satisfes the condition |e f (0) − e f (1)| � 0. Case 2: when n is odd. Hence, according to Case 1 and Case 2, the graph G is a power cordial graph.

Some NP-Complete Problems of Power Cordial Graphs
Most of the decision problems are classifed into four types: P, NP, NP-complete, and NP-hard. Although one can verify the solution quickly for an NP-complete problem, but there does not exist efcient algorithm to reach the solution quickly. Many graph theoretic problems such as to fnd the domination set, to assign proper vertex coloring, and to fnd the longest path in the graph are NP-complete. More details are available in [7]. Acharya et al. [8] have derived that every graph can be embedded as an induced subgraph of a graceful graph, thus showing the impossibility of obtaining any forbidden subgraph characterization for graceful graphs. On the same line, Acharya et al. [9,10], Sethuraman et al. [11], Germina and Ajitha [12], Ichishima et al. [13], Rao and Sahoo [14], Vaidya and Barasara [15,16], Vaidya and Vihol [17], and Anandavally et al. [18] have discussed embedding and NP-complete problems in the context of various graph labeling schemes.
In this section, we study some NP-complete problems in the context of power cordial labeling.
Theorem 15. Te problem of deciding whether the chromatic number χ(G) ≤ k, where k ≥ 3, is NP-complete even for a power cordial graph.
Proof. Let G be a graph with the chromatic number χ(G) ≥ 3. Let H be the power cordial supergraph constructed in Teorem 3, which contains G as an induced subgraph. Ten, χ(H) ≥ χ(G). Since the problem of deciding whether the chromatic number χ(G) ≤ k, where k ≥ 3, is NP-complete by [7]. It follows that deciding whether the chromatic number χ(H) ≤ k, where k ≥ 3, is NP-complete even for power cordial graphs. Hence, the result.  Proof. Since the problem of deciding whether the clique number of a graph ω(G) ≥ k is NP-complete by [7] and ω(H) ≥ ω(G) for the supergraph H constructed in Teorem 3 is power cordial. Hence, the result.
□ Theorem 17. Te problem of deciding whether the domination number (total domination number) of a graph G is less than or equal to k is NP-complete even when restricted to the power cordial graph.
Proof. Since the problem of deciding whether the domination number (total domination number) of a graph G is less than or equal to k is NP-complete, as reported in [7]. Te supergraph H constructed in Teorem 3 is power cordial, and its domination number is greater than or equal to the domination number of G. Hence, the result.

Conclusions
In this paper, we have derived how any nonpower cordial graph can be embedded in a power cordial graph and obtained some characterizations for a power cordial graph. We have also discussed some NP-complete problems for power cordial labeling, which rules out the possibility of forbidden subgraph characterization for power cordial labeling. To obtain similar results for other graph labeling schemes is an open area of research.

Data Availability
No data were used to support the fndings of this study.

Disclosure
Te present work is a part of the research work conducted under the minor research project no. HNGU/UGC/5658/ 2023, dated: 4th January, 2023, of Hemchandracharya North Gujarat University, Patan (Gujarat), INDIA.