Cartwright and Harary considered graphs in which vertices represent persons and the edges represent symmetric dyadic relations amongst persons each of which designated as being positive or negative according to whether the nature of the relationship is positive (friendly, like, etc.) or negative (hostile, dislike, etc.). Such a network S is called a signed graph. Signed graphs are much studied in literature because of their extensive use in modeling a variety socio-psychological process and also because of their interesting connections with many classical mathematical systems

Reddy, Siva Kota

Vijay, S.

English

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International J.Math. Combin. Vol.3 (2010), 11-16Total Minimal Dominating Signed GraphP.Siva Kota Reddy(Department of Mathematics, Acharya Institute of Technology, Bangalore-560 090, India)S. Vijay(Department of Mathematics, Govt. First Grade College, Kadur, Chikkamangalore 577 548, India)Email: pskreddy@acharya.ac.inAbstract: A Smarandachely k-signed graph (Smarandachely k-marked graph) is an orderedpair S = (G,σ) (S = (G,µ)) where G = (V, E) is a graph called underlying graph of S andσ : E → (e1, e2, ..., ek) (µ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}.Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is calledabbreviated a signed graph or a marked graph. In this paper, we define the total minimaldominating signed graph Mt(S) = (Mt(G), σ) of a given signed digraph S = (G,σ) andoffer a structural characterization of total minimal dominating signed graphs. Further, wecharacterize signed graphs S for which S ∼ Mt(S) and L(S) ∼ Mt(S), where ∼ denotesswitching equivalence and Mt(S) and L(S) are denotes total minimal dominating signedgraph and line signed graph of S respectively.Key Words: Smarandachely k-signed graphs, Smarandachely k-marked graphs, signedgraphs, marked graphs, balance, switching, total minimal dominating signed graph, linesigned graphs, negation.AMS(2000): 05C22§1. IntroductionUnless mentioned or defined otherwise, for all terminology and notion in graph theory thereader is refer to [8]. We consider only finite, simple graphs free from self-loops.A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pairS = (G, σ) (S = (G,µ)) where G = (V,E) is a graph called underlying graph of S andσ : E → (e1, e2, ..., ek) (µ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}. Particu-larly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviateda signed graph or a marked graph. Cartwright and Harary [5] considered graphs in which ver-tices represent persons and the edges represent symmetric dyadic relations amongst personseach of which designated as being positive or negative according to whether the nature of the1Received July 20, 2010. Accepted September 3, 2010.12 P.Siva Kota Reddy and S. Vijayrelationship is positive (friendly, like, etc.) or negative (hostile, dislike, etc.). Such a networkS is called a signed graph (Chartrand [6]; Harary et al. [11]).Signed graphs are much studied in literature because of their extensive use in modeling avariety socio-psychological process (e.g., see Katai and Iwai [13], Roberts [15] and Roberts andXu [16]) and also because of their interesting connections with many classical mathematicalsystems (Zaslavsky [22]).A cycle in a signed graph S is said to be positive if the product of signs of its edges ispositive. A cycle which is not positive is said to be negative. A signed graph is then said to bebalanced if every cycle in it is positive (Harary [9]). Harary and Kabell [22] developed a simplealgorithm to detect balance in signed graphs as also enumerated them.A marking of S is a function µ : V (G) → {+,−}; A signed graph S together with amarking µ is denoted by Sµ. Given a signed graph S one can easily define a marking µ of S asfollows: For any vertex v ∈ V (S),µ(v) =∏uv∈E(S)σ(uv),the marking µ of S is called canonical marking of S.The following characterization of balanced signed graphs is well known.Theorem 1(E. Sampathkumar [17]) A signed graph S = (G, σ) is balanced if, and only if,there exists a marking µ of its vertices such that each edge uv in S satisfies σ(uv) = µ(u)µ(v).The idea of switching a signed graph was introduced by Abelson and Rosenberg [1] inconnection with structural analysis of marking µ of a signed graph S. Switching S with respectto a marking µ is the operation of changing the sign of every edge of S to its opposite wheneverits end vertices are of opposite signs. The signed graph obtained in this way is denoted bySµ(S) and is called µ-switched signed graph or just switched signed graph. Two signed graphsS1 = (G, σ) and S2 = (G′, σ′) are said to be isomorphic, written as S1 ∼= S2 if there existsa graph isomorphism f : G → G′ (that is a bijection f : V (G) → V (G′) such that if uv isan edge in G then f(u)f(v) is an edge in G′) such that for any edge e ∈ G, σ(e) = σ′(f(e)).Further a signed graph S1 = (G, σ) switches to a signed graph S2 = (G′, σ′) (or that S1 and S2are switching equivalent) written S1 ∼ S2, whenever there exists a marking µ of S1 such thatSµ(S1) ∼= S2. Note that S1 ∼ S2 implies that G ∼= G′, since the definition of switching doesnot involve change of adjacencies in the underlying graphs of the respective signed graphs.Two signed graphs S1 = (G, σ) and S2 = (G′, σ′) are said to be weakly isomorphic (see[20]) or cycle isomorphic (see [21]) if there exists an isomorphism φ : G → G′ such that thesign of every cycle Z in S1 equals to the sign of φ(Z) in S2. The following result is well known(See [21]):Theorem(T. Zaslavsky [21]) Two signed graphs S1 and S2 with the same underlying graph areswitching equivalent if, and only if, they are cycle isomorphic.Total Minimal Dominating Signed Graph 13§2. Total Minimal Dominating Signed GraphThe total minimal dominating graphMt(G) of a graph G is the intersection graph on the familyof all total minimal dominating sets of vertices in G. This concept was introduced by Kulli andIyer [14].We now extend the notion of Mt(G) to the realm of signed graphs. The total minimaldominating signed graph Mt(S) of a signed graph S = (G, σ) is a signed graph whose underlyinggraph is Mt(G) and sign of any edge uv is Mt(S) is µ(u)µ(v), where µ is the canonical markingof S. Further, a signed graph S = (G, σ) is called total minimal dominating signed graph, ifS ∼= Mt(S′) for some signed graph S′. The following result restricts the class of total minimaldominating signed graphs.Theorem 3 For any signed graph S = (G, σ), its total minimal dominating signed graph Mt(S)is balanced.Proof Since sign of any edge uv is Mt(S) is µ(u)µ(v), where µ is the canonical marking ofS, by Theorem 1, Mt(S) is balanced. For any positive integer k, the kth iterated total minimal dominating signed graph,Mkt (S)of S is defined as follows:M0t (S) = S, Mkt (S) =Mt(Mk−1t (S))Corollary 4 For any signed graph S = (G, σ) and for any positive integer k,Mkt (S) is balanced.The following result characterizes signed graphs which are total minimal dominating signedgraphs.Theorem 5 A signed graph S = (G, σ) is a total minimal dominating signed graph if, and onlyif, S is balanced signed graph and its underlying digraph G is a total minimal dominating graph.Proof Suppose that S is total minimal dominating signed graph. Then there exists a signedgraph S′ = (G′, σ′) such that Mt(S′) ∼= S. Hence by definition Mt(G) ∼= G′ and by Theorem3, S is balanced.Conversely, suppose that S = (G, σ) is balanced and G is total minimal dominating graph.That is there exists a graph G′ such that Mt(G′) ∼= G. Since S is balanced by Theorem 1,there exists a marking µ of vertices of S such that for any edge uv ∈ G, σ(uv) = µ(u)µ(v).Also since G ∼= Mt(G′), vertices in G are in one-to-one correspondence with the edges of G′.Now consider the signed graph S′ = (G′, σ′), where for any edge e′ in G′ to be the marking onthe corresponding vertex in G. Then clearly Mt(S′) ∼= S and so S is total minimal dominatinggraph. In [3], the authors proved the following for a graph G its total minimal dominating graphMt(G) is isomorphic to G then G is either C3 or C4. Analogously we have the following.Theorem 6 For any signed graph S = (G, σ), S ∼ Mt(S) if, and only if, G is isomorphic toeither C3 or C4 and S is balanced.14 P.Siva Kota Reddy and S. VijayProof Suppose S ∼Mt(S). This implies, G ∼=Mt(G) and hence by the above observationwe see that the graph G must be isomorphic to either C3 or C4. Now, if S is any signed graph onany one of these graphs, Theorem 3 implies thatMt(S) is balanced and hence if S is unbalancedits Mt(S) being balanced cannot be switching equivalent to S in accordance with Theorem 2.Therefore, S must be balanced.Conversely, suppose that S is balanced signed graph on C3 or C4. Then, since Mt(S) isbalanced as per Theorem 3, the result follows from Theorem 2 again. Behzad and Chartrand [4] introduced the notion of line signed graph L(S) of a given signedgraph S as follows: Given a signed graph S = (G, σ) its line signed graph L(S) = (L(G), σ′) isthe signed graph whose underlying graph is L(G), the line graph of G, where for any edge eiejin L(S), σ′(eiej) is negative if, and only if, both ei and ej are adjacent negative edges in S.Another notion of line signed graph introduced in [7], is as follows: The line signed graph of asigned graph S = (G, σ) is a signed graph L(S) = (L(G), σ′), where for any edge ee′ in L(S),σ′(ee′) = σ(e)σ(e′). In this paper, we follow the notion of line signed graph defined by M. K.Gill [7] (See also E. Sampathkumar et al. [18,19]).Theorem 7(M. Acharya [2]) For any signed graph S = (G, σ), its line signed graph L(S) =(L(G), σ′) is balanced.We now characterize signed graphs whose total minimal dominating signed graphs and itsline signed graphs are switching equivalent. In the case of graphs the following result is due toKulli and Iyer [14].Theorem 8(Kulli and Iyer [14]) If G is a (p− 2)-regular graph then, Mt(G) ∼= L(G).Theorem 9 For any signed graph S = (G, σ), Mt(S) ∼ L(S), if, and only if, G is (p − 2)-regular.Proof Suppose Mt(S) ∼ L(S). This implies, Mt(G) ∼= L(G) and hence by Theorem 8, wesee that the graph G must be (p− 2)-regular.Conversely, suppose that G is (p − 2)-regular. Then Mt(G) ∼= L(G) by Theorem 8. Nowif S is signed graph with (p− 2)-regular, then by Theorem 3 and Theorem 7, Mt(S) and L(S)are balanced and hence, the result follows from Theorem 2. The notion of negation η(S) of a given signed graph S defined in [10] as follows:η(S) has the same underlying graph as that of S with the sign of each edge opposite tothat given to it in S. However, this definition does not say anything about what to do withnonadjacent pairs of vertices in S while applying the unary operator η(.) of taking the negationof S.Theorem 6 provides easy solutions to two other signed graph switching equivalence rela-tions, which are given in the following results.Corollary 10 For any signed graph S = (G, σ), Mt(η(S)) ∼Mt(S).Total Minimal Dominating Signed Graph 15Corollary 11 For any signed graph S = (G, σ), η(S) ∼ Mt(S) if, and only if, S is anunbalanced signed graph and G = C3.For a signed graph S = (G, σ), the Mt(S) is balanced (Theorem 3). We now examine, theconditions under which negation η(S) of Mt(S) is balanced.Corollary 12 Let S = (G, σ) be a signed graph. If Mt(G) is bipartite then η(Mt(S)) isbalanced.Proof Since, by Theorem 3 Mt(S) is balanced, if each cycle C in Mt(S) contains evennumber of negative edges. Also, since Mt(G) is bipartite, all cycles have even length; thus, thenumber of positive edges on any cycle C in Mt(S) is also even. Hence η(Mt(S)) is balanced.AcknowledgementThe first author very much thankful to Sri. B. Premnath Reddy, Chairman, Acharya Institutes,for his constant support and encouragement for R & D.References[1] R. P. Abelson and M. J. Rosenberg, Symoblic psychologic:A model of attitudinal cognition,Behav. Sci., 3 (1958), 1-13.[2] M. Acharya, x-Line sigraph of a sigraph, J. Combin. Math. Combin. Comput., 69(2009),103-111.[3] B. Basvanagoud and S. M. Hosamani, Miscellaneous properties of the total minimal dom-inating graph, Journal of Analysis and Computation, to appear.[4] M. Behzad and G. T. Chartrand, Line-coloring of signed graphs, Elemente der Mathematik,24(3) (1969), 49-52.[5] D.W. Cartwright and F. Harary, Structural balance: A generalization of Heider’s Theory,Psych. Rev., 63(1956), 277-293.[6] G.T. Chartrand, Graphs as Mathematical Models, Prindle, Weber & Schmidt, Inc., Boston,Massachusetts 1977.[7] M. K. Gill, Contributions to some topics in graph theory and its applications, Ph.D. thesis,The Indian Institute of Technology, Bombay, 1983.[8] F. Harary, Graph Theory, Addison-Wesley Publishing Co., 1969.[9] F. Harary, On the notion of balance of a signed graph,Michigan Math. J., 2(1953), 143-146.[10] F. Harary, Structural duality, Behav. Sci., 2(4) (1957), 255-265.[11] F. Harary, R.Z. Norman and D.W. Cartwright, Structural models: An introduction to thetheory of directed graphs, Wiley Inter-Science, Inc., New York, 1965.[12] F. Harary and J.A. Kabell, Counting balanced signed graphs using marked graphs, Proc.Edinburgh Math. Soc., 24 (2)(1981), 99-104.[13] O. Katai and S. Iwai, Studies on the balancing, the minimal balancing, and the minimumbalancing processes for social groups with planar and nonplanar graph structures, J. Math.16 P.Siva Kota Reddy and S. VijayPsychol., 18(1978), 140-176.[14] V. R. Kulli and R. R. Iyer, The total minimal dominating graph of a graph-Preprint.[15] F.S. Roberts, Graph Theory and its Applications to Problems of Society, SIAM, Philadel-phia, PA, USA, 1978.[16] F.S. Roberts and Shaoji Xu, Characterizations of consistent marked graphs, Discrete Ap-plied Mathematics, 127(2003), 357- 371.[17] E. Sampathkumar, Point signed and line signed graphs, Nat. Acad. Sci. Letters, 7(3)(1984), 91-93.[18] E. Sampathkumar, P. Siva Kota Reddy, and M. S. Subramanya, The Line n-sigraph of asymmetric n-sigraph, Southeast Asain Bull. Math., 34(4) (2010), to appear.[19] E. Sampathkumar, M. S. Subramanya and P. Siva Kota Reddy, Characterization of LineSidigraphs, Southeast Asian Bull. Math., to appear.[20] T. Soza´nsky, Enueration of weak isomorphism classes of signed graphs, J. Graph Theory,4(2)(1980), 127-144.[21] T. Zaslavsky, Signed Graphs, Discrete Appl. Math., 4(1)(1982), 47-74.[22] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and its allied areas,Electronic J. Combin., 8(1)(1998), Dynamic Surveys (1999), No. DS8.

A mathematical bibliography of signed and gain graphs and its allied areas,

Characterizations of consistent marked graphs,

Contributions to some topics in graph theory and its applications,

Counting balanced signed graphs using marked graphs,

Enueration of weak isomorphism classes of signed graphs,

Graph Theory and its Applications to Problems of Society,

Graph Theory,

Graphs as Mathematical Models,

Line-coloring of signed graphs,

Miscellaneous properties of the total minimal dominating graph,

On the notion of balance of a signed graph,Michigan

Point signed and line signed graphs,

Signed Graphs,

Structural balance: A generalization of Heider’s Theory,

Structural duality,

Structural models: An introduction to the theory of directed graphs,

Studies on the balancing, the minimal balancing, and the minimum balancing processes for social groups with planar and nonplanar graph structures,

Total Minimal Dominating Signed Graph

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