^{1}Dept. of Mathematics, Acharya Institute of Technology, Bangalore-560 090, India.

^{2}Professor and Head Dept. of Mathematics Acharya Institute of Technology Bangalore-560 090 India

^{3}Assistant Professor Dept.of Mathematics Acharya Institute of Technology Bangalore-560 090 India.

Abstract

In this paper, we define the common minimal common neighborhood dominating signed graph (or common minimal $CN$-dominating signed graph) of a given signed graph and offer a structural characterization of common minimal $CN$-dominating signed graphs. In the sequel, we also obtained switching equivalence
characterization: $\overline{\Sigma} \sim CMCN(\Sigma)$, where $\overline{\Sigma}$ and $CMCN(\Sigma)$ are complementary signed graph and common minimal $CN$-signed graph of $\Sigma$ respectively.

R. P. Abelson and M. J. Rosenberg (1958). Symoblic psychologic: A model of
attitudinal cognition. Behav. Sci.. 3, 1-13

2

A. Alwardi, N. D. Soner and K. Ebadi (2011). On the common neighbourhood domination
number. J. Comp. $&$ Math. Sci.. 2 (3), 547-556

3

A. Alwardi and N. D. Soner (2012). Minimal, vertex minimal and commonality minimal CN-dominating
graphs. Trans. Comb.. 1 (1), 21-29

4

C. Berge (1962). Theory of Graphs and its Applications. Methuen, London.

5

E. J. Cockayne and S. T. Hedetniemi (1977). Towards a theory of domination in graphs. Networks. 7, 247-261

6

C. F. De Jaenisch (1862). Applications de l’Analyse mathematique an Jen des Echecs.

7

D. Easley and J. Kleinberg (2010). Networks, Crowds, and Markets: Reasoning About
a Highly Connected World. Cambridge University Press.

8

F. Harary (1969). Graph Theory. Addison-Wesley Publishing Co..

9

F. Harary (1953). On the notion of balance of a signed graph. Michigan Math. J.. 2, 143-146

10

F. Harary (1957). Structural duality. Behavioral Sci.. 2 (4), 255-265

11

O. Ore (1962). Theory of Graphs. Amer. Math. Soc. Colloq. Publ.. 38

12

R. Rangarajan and P. Siva Kota Reddy (2010). The edge $C_4$ signed graph of a signed graph. Southeast Asian Bull. Math.. 34 (6), 1077-1082

13

R. Rangarajan, M. S. Subramanya and P. Siva Kota Reddy (2012). Neighborhood signed graphs. Southeast Asian Bull. Math.. 36 (3), 389-397

14

W. W. Rouse Ball (1892). Mathematical Recreation and Problems of Past and Present Times.

15

E. Sampathkumar (1984). Point signed and line signed graphs. Nat.
Nat. Acad. Sci. Lett.. 7 (3), 91-93

16

E. Sampathkumar, P. Siva Kota Reddy and M. S.
Subramanya (2010). Directionally $n$-signed graphs. Ramanujan Math. Soc., Lecture Notes
Series (Proc. Int. Conf. ICDM 2008). 13, 155-162

17

E. Sampathkumar, P. Siva Kota Reddy and M. S. Subramanya (2009). Directionally $n$-signed graphs-II. Int. J. Math. Comb.. 4, 89-98

18

E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy (2011). Characterization of line sidigraphs. Southeast Asian Bull. Math.. 35 (2), 297-304

19

P. Siva Kota Reddy and M. S. Subramanya (2009). Note on path signed graphs. Notes on Number Theory and Discrete Mathematics. 15 (4), 1-6

20

P. Siva Kota Reddy, S. Vijay and V. Lokesha (2009). $n^{th}$ Power signed graphs. Proc. Jangjeon Math. Soc.. 12 (3), 307-313

21

P. Siva Kota Reddy (2010). $t$-Path Sigraphs. Tamsui Oxf. J. Math. Sci.. 26 (4), 433-441

22

P. Siva Kota Reddy, E. Sampathkumar and M. S. Subramanya (2010). Common-edge signed graph of a
signed graph. J. Indones. Math. Soc.. 16 (2), 105-112

23

P. Siva Kota Reddy, B. Prashanth and T. R. Vasanth Kumar (2011). Antipodal signed directed
Graphs. Advn. Stud. Contemp. Math.. 21 (4), 355-360

24

P. Siva Kota Reddy and B. Prashanth (2012). The Common Minimal Dominating Signed
Graph. Trans. Comb.. 1 (3), 39-46

25

P. Siva Kota Reddy and B. Prashanth (2012). $mathcal{S}$-Antipodal signed graphs. Tamsui Oxford J. of Inf. Math. Sciences. 28 (2), 165-174

26

P. Siva Kota Reddy and S. Vijay (2012). The super line signed graph $mathcal{L}_r(S)$ of a signed Graph. Southeast Asian Bulletin of Mathematics. 36 (6), 875-882

27

P. Siva Kota Reddy and U. K. Misra (2012). Common Minimal Equitable Dominating Signed
Graphs. Notes on Number Theory and Discrete Mathematics. 18 (4), 40-46

28

P. Siva Kota Reddy and U. K. Misra (2013). The Equitable Associate Signed
Graphs. Bull. Int. Math. Virtual Inst.. 3 (1), 15-20

29

P. Siva Kota Reddy, U. K. Misra and P. N. Samanta (2013). The Minimal Equitable Dominating Signed
Graphs. Bull. of Pure $&$ Appl. Math., to appear.

30

P. Siva Kota Reddy and B. Prashanth (2012). Note on Minimal Dominating Signed
Graphs. Bull. of Pure $&$ Appl. Math., to appear.

31

T. Soz$acute{a}$nsky (1980). Enueration of weak isomorphism
classes of signed graphs. J. Graph Theory. 4 (2), 127-144

32

A. M. Yaglom and I. M. Yaglom (1964). Challenging mathematical problems with elementary solutions. Combinatorial Analysis and Probability Theory. 1

33

T. Zaslavsky (1982). Signed graphs. Discrete Appl. Math.. 4 (1), 47-74

34

T. Zaslavsky (1998). A mathematical bibliography of signed and gain graphs and its allied areas. Electron. J. Combin., Dynamic Surveys, no. DS8. 8 (1)