Gallai Signed Graphs and Anti-Gallai Signed Graphs K.V. Madhusudhan 1 , Sashi Kanth Reddy 2 and P. Siva Kota Reddy 3 1 Department of Mathematics, ATME College of Engineering, Mysore-570 028, India Email: kvmadhu13@gmail.com 2 Department of Computer Science, AIMS Institutes, Bangalore, India Email: sashikanth@theaims.ac.in 3 Department of Mathematics, JSS Science and Technology University, Mysuru-570 006, INDIA Email: pskreddy@jssstuniv.in; pskreddy@sjce.ac.in Abstract— In this paper we introduced the new notions Gallai and anti-Gallai signed graph of a signed graph and its properties are obtained. Also, we obtained the structural characterizations of these notions. Further, we presented some interesting switching equivalent characterizations. Index Terms— Mathematics Subject Classification: 05C22, 05C12, Signed graphs, Balance, Switching, Gallai Signed graph, Anti-Gallai Signed Graph. I. INTRODUCTION For standard terminology and notation in graph theory we refer Harary [4] and Zaslavsky [27] for signed graphs. Throughout the text, we consider finite, undirected graph with no loops or multiple edges. The Gallai Graph ℒ(ܩ) of a Graph G = (V, E) is the Graph whose Vertex Set ൫ℒ (ܩ)൯ = ܧ(ܩ); two distinct vertices e 1 and e 2 are adjacent in ℒ(ܩ) if e 1 and e 2 are incident in G, but do not span a triangle in G (see [6]). In fact, this concept was introduced by Gallai [3] in his examination of comparability Graphs and this notion was recommended by Sun [25]. The author Sun wasted Gallai Graphs ℒ (ܩ) to characterize an amusing class of perfect Graphs. Gallai Graphs are also wasted in the Polynomial Time Algorithm to determinate complete Bipartite K (1,3) free perfect Graphs by the authors Chvatal and Sbihi [2]. The Anti-Gallai Graph ℒ(ܩ) of a Graph G = (V, E) is the Graph whose Vertex Set ൫ℒ(ܩ)൯ = ܧ(ܩ); two distinct vertices e 1 and e 2 are adjacent in ℒ (ܩ) if e 1 and e 2 are incident in G and lie on a triangle in G (see [6]). Equivalently, the Anti-Gallai Graph ℒ(ܩ) is the complement of Gallai Graphℒ(ܩ) in the line Graph L (G). We can easily observe that the Gallai Graphs ℒ (ܩ) and Anti-Gallai Graphs ℒ(ܩ) are the spanning subgraphs of the Line Graph L (G). A signed graph  =( ܩ, ߪ) is graph with each edge labeled by a sign (i.e.,+ or - ). Clearly, S is a mapping form E (G) to {+, -} (e.g., see Harary [5], Sampathkumar [11], Zaslavsky [26, 27]): the set {+, -} may either treated simply as a set of signs called colors, when ߪis regraded as a 2-edge coloring of G or as the involutory Group {- 1, +1} demarcated by a multiplication table in which ߪis preserved as a estimation of the edges of S. For given Signed Graph  =(Γ, ߪ), the edge set E(S) could be classified as ܧ()= ܧ ା () ܧ∪  (), where E + (S)(E - (S)) denote the set of all edges marked by + (-). The Sign of a cycle in S is the multiplication of the Signs of all edges on the Cycle. For more new notions on signed graphs refer the papers (See [8-10, 13-24]). Grenze ID: 01.GIJET.9.1.676 © Grenze Scientific Society, 2023 Grenze International Journal of Engineering and Technology, Jan Issue