International Journal of Computer Applications (0975 – 8887) Volume 93 – No 10, May 2014 19 An Optimal Algorithm to Detect Balancing in Common- edge Sigraph Deepa Sinha South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi – 110 021. Anshu Sethi Center for Mathematical Sciences, Banasthali University, Banasthali – 304 022 ABSTRACT A signed graph (or sigraph in short) S is a graph G in which each edge x carries a value s(x) ∈ {+1, −1} called its sign denoted specially as S = (G, s). Given a sigraph S, a new sigraph C E (S), called the common-edge sigraph of S is that sigraph whose vertex-set is the set of pairs of adjacent edges in S and two vertices of C E (S) are adjacent if the corresponding pairs of adjacent edges of S have exactly one edge in common, and the sign of the edge is the sign of the common edge. If all the edges of the sigraph S carry + sign then S is actually a graph and the corresponding common- edge sigraph is termed as the common-edge graph. In this paper, algorithms are defined to obtain a common-edge sigraph and detect whether it is balanced or not in O(n 3 ) steps which will be optimal in nature. Keywords Algorithm, sigraph, common-edge graph, common-edge sigraph, balanced signed graph. 1. INTRODUCTION For standard terminology and notation in graph theory, except for those that are specifically defined here, the reader is referred to West [33] and for algorithms, refer to Coreman [12]. Throughout the text, finite, undirected graph with no loops or multiple edges are considered. A graph having n vertices and e edges; is denoted by (n, e) where n is called the order and e is called the size of G. In computers, any graph G is observed as network by computer scientist where vertices are taken to be nodes and edges to be taken as links. In the spirit of a study of graph-valued functions, obtaining the line graph L(G) of a given graph G = (V, E) may be treated as a mapping L that operates on G to give rise to L(G) as the graph whose vertices are the edges of G with two of these vertices joined to each other (or, adjacent) whenever the edges of G they represent have a common vertex in G or equivalently the two edges form a P 3 in G. H is called line graph if and only if ∃ a graph G such that H ≅ L(G). . Broersma and Hoede [9] defined in general path graphs P k (G) of G for any positive integer k as follows: P k (G) has for its vertex-set the set P k G of all distinct paths in G having k vertices, and two vertices in P k (G) are adjacent if they represent two paths P, Q ∈ P k G whose union forms either a path P k+1 or a cycle C k in G. Some improvement of their paper was subsequently given by [27, 7, 28]. Much earlier, making independently the same observation as above on the formation of a line graph L(G) of a given graph G, Kulli [32] had defined the common-edge graph C E (G) of G as the intersection graph of the family P 3 (G) of 2-paths (i.e., paths of length two) each member of which is treated as a set of edges of the corresponding 2-path; as shown by him, it is not difficult to see that C E (G) ≅ L 2 (G) for any isolate-free graph G, where L(G) := L 1 (G) and L t (G) denotes the t-th iterated line graph of G for any integer t ≥ 2. The notion of L(G) has been extended to the realm of signed graph (or sigraph in short) [8]. As in [9] (also see, [7]) by a sigraph S we mean a 2 graph G = (V, E) called the underlying graph of S and denoted by S u , in which each edge x carries a value s(x) ∈ {+1, −1} called its sign; an edge x is positive or negative according to whether s(x) = +1 or s(x) = −1. The set of positive edges of S is denoted by E + (S) and E − (S) = E(G) − E + (S) is the set of negative edges of S. Graphs themselves regarded as sigraphs in which every edge is positive. Given a graph G, let ᵩ(S) denote the set of all sigraphs whose underlying graph is G. In general, a subgraph Sꞌ of a sigraph S is said to be all-positive(all-negative) if all the edges of Sꞌ are positive (negative). A sigraph is said to be homogeneous if it is either all-positive or all-negative and heterogeneous otherwise. Cliques are defined as the complete subgraphs of the graph. For a sigraph S, Behzad and Chartrand [8] defines its common-edge sigraph, C E (S) as the sigraph whose vertex-set is the set of pairs of adjacent edges in S and two vertices of C E (S) are adjacent if the corresponding pairs of adjacent edges of S have exactly one edge in common, with the same sign as that of common edge.