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International Journal of Pure and Applied Mathematics Volume 101 No. 6 2015, 939-948 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: A http://www.ijpam.eu P ijpam.eu SSP STRUCTURE OF SOME GRAPH CLASSES R. Mary Jeya Jothi1, A. Amutha2 1,2Department of Mathematics Sathyabama University Chennai, 119, INDIA Abstract: A Graph G is Super Strongly Perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. Strongly perfect, complete bipartite graph, etc., are some of the most important classes of super strongly perfect graphs. Here, we analyse the other classes of super strongly perfect graphs like trivially perfect graphs and very strongly perfect graphs. We investigate the structure of super strongly perfect graph in friendship graphs. Also, we discuss the maximal cliques, colourability and dominating sets in friendship graphs. AMS Subject Classification: 05C75 Key Words: super strongly perfect graph, trivially perfect graph, very strongly perfect graph and friendship graph 1. Introduction Super strongly perfect graph is defined by B.D. Acharya and its characterization has been given as an open problem in 2006 [5]. Many analysation on super strongly perfect graph have been done by us in our previous research work, (i.e.,) we have investigated the classes of super strongly perfect graphs like strongly c 2015 Academic Publications, Ltd. Received: March 12, 2015 url: www.acadpubl.eu 940 R.M.J. Jothi, A. Amutha perfect, complete bipartite graph, etc., [3, 4]. In this paper we have discussed some other classes of super strongly perfect graphs like trivially perfect graph, very strongly perfect graph and friendship graph. Also, we analyse the maximal cliques, colourability and minimal dominating sets in friendship graph. In this paper, graphs are finite, undirected and simple, that is, they have no loops or multiple edges. Let G be a graph. A Clique in G is a set X ⊆ V (G) of pair wise adjacent vertices. A subset D of V (G) is called a Dominating set if every vertex in V \D is adjacent to at least one vertex in D. A subset S of V is said to be a Minimal Dominating Set if S − {u} is not a dominating set for any u ∈ S. By vertex colouring or colouring of a graph G, we mean assigning different colours to the vertices of the graph such that no two adjacent vertices get the same colour. The Chromatic number of a graph G is the minimum number of colours that is required to colour G. A graph G = (V, E) is Strongly Perfect (SP) if every induced sub graphs H of G contains a stable set that meets all the maximal cliques of H. 2. Super Strongly Perfect Graph (SSP) A Graph G = (V, E) is super strongly perfect if every induced sub graph H of G possesses a minimal dominating set that meets all maximal cliques of H. The following Figures 1 and 2 illustrate the structure of the super strongly perfect and non super strongly perfect graphs. Example 1. Figure 1: Super strongly perfect graph Here, {3, 6} is a minimal dominating set which meets all maximal cliques of G. Example 2. Here, {1, 3, 6, 8} is a minimal dominating set which does not meet all max- imal cliques of G. SSPSTRUCTUREOFSOMEGRAPHCLASSES 941 Figure 2: Non-super strongly perfect graph Theorem 3. [4] Every strongly perfect graph is super strongly perfect. Theorem 4. [3] Let G be a graph with at least one maximal clique Kn, n = 2, 3,... then G is super strongly perfect if and only if it is n-colourable. 3. Helm Graph The Helm graph Hn is the graph obtained from an n-wheel graph by adjoining a pendant edge at each vertex of the cycle that is helm graph Hn is obtained by attaching a single edge and vertex to each vertex of the outer cycle of a wheel graph Wn. Figure 3 illustrates the helm graph H7 which is Super Strongly Perfect. Example 5. Figure 3: H7 Theorem 6. Every helm graph Hn where n is odd, is Super Strongly Perfect. 942 R.M.J. Jothi, A. Amutha Proof. Let G be a helm graph Hn where n is odd. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the outer cycle. Since every odd wheel graph is Super Strongly Perfect, if we join a pendent edge at each vertex of the cycle of the odd wheel then again the graph is Super Strongly Perfect with a minimal dominating set of cardinality n vertices (i.e.,) the n − 1 non adjacent vertices from the n − 1 pendent edges and a middle vertex from the wheel graph. Hence G is Super Strongly Perfect. Theorem 7. Every helm graph Hn where n is even, is non-Super Strongly Perfect. Proof. Let G be a helm graph Hn where n is even. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the cycle. Since every even wheel graph is non-Super Strongly Perfect, if we join a pendent edge at each vertex of the cycle of the even wheel, then again the graph is non-Super Strongly Perfect. Hence G is non-Super Strongly Perfect. Theorem 8. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G contains (n − 1) maximal cliques K3. Proof. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ G is obtained from an n-wheel graph by adjoining a pendant edge at each vertex of the outer cycle Cn−1. ⇒ Any two adjacent vertices from Cn−1 with a centre single vertex give an induced K3. Since G has n − 1 such vertices, G has (n − 1) maximal cliques K3. Hence proved. Theorem 9. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G is 3-colourable. Proof. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ by the previous theorem, G has (n − 1) maximal cliques on 3 vertices. SSPSTRUCTUREOFSOMEGRAPHCLASSES 943 ⇒ by the Theorem 4, G is 3-colourable. Hence proved. Theorem 10. Let G be a helm graph with odd number of vertices, which is Super Strongly Perfect, then G contains a minimal dominating set of cardinality n where n is the number of vertices. Proof. Let G be a Helm graph with odd number of vertices, which is Super Strongly Perfect. ⇒ G is constructed by an n-wheel graph by joining a pendent edge at each vertex of the outer cycle. ⇒ The n−1 non adjacent pendent edges of the vertices and a middle vertex from the wheel graph gives a minimal dominating set which meet all maximal cliques K3. ⇒ G has a minimal dominating set of cardinality n. Hence proved. 4. Friendship Graph A Friendship graph Fk on 2k+1 vertices is the graph obtained by taking k copies of the cycle graph C3 with a vertex in common, that is, the graph of k triangles intersecting in a single vertex. Hence every friendship graph contains a single vertex which is adjacent to all the remaining vertices. It is unit distance with girth 3, diameter 2 and radius 1. The friendship theorem of Paul Erdos, et.al. [1], states that the finite graphs with the property that every two vertices have exactly one neighbour in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly a friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs with the same cardinality that have this property. Figure 4 illustrates the friendship graph F6 which is super strongly perfect. Example 11. Here {1} is a minimal dominating set which meets all the maximal cliques of G. Theorem 12. Every friendship graph is super strongly perfect. 944 R.M.J. Jothi, A. Amutha Figure 4: F6 Proof. Let G be a friendship graph. ⇒ Let S = {v} be the minimal dominating set of cardinality 1. ⇒ v is adjacent to all the remaining vertices of G. ⇒ v meets all maximal cliques of G. ⇒ G is super strongly perfect. Hence proved. Observation 13. Let G be a friendship graph Fk. Then G is 3-colourable. Proof. Let G be a friendship graph Fk. ⇒ G has k maximal cliques on 3 vertices. By the Theorem 4, G is 3-colourable. Hence proved. Observation 14. Let G be a friendship graph Fk. Then G has a minimal dominating set of maximum cardinality k. Proof. Let G be a friendship graph Fk. ⇒ G has k maximal cliques on 3 vertices. Since G has k copies of K3, there exists a minimal dominating set of cardi- nality k which meet all K3. ⇒ G has a minimal dominating set of maximum cardinality k. Hence proved. 5. Trivially Perfect Graph (TP) A Graph G is trivially perfect if in every induced sub graph H of G the size of the largest independent set equals the number of maximal cliques. Figure 5 SSPSTRUCTUREOFSOMEGRAPHCLASSES 945 illustrates the trivially perfect graph which is super strongly perfect. Example 15. Figure 5: Trivially perfect graph Here, the cardinality of the largest independent set is 2 and also the graph has 2 maximal cliques of size 4. Theorem 16. Every trivially perfect graph is strongly perfect. Proof. Let G be a trivially perfect graph.
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