The decay number and the maximum genus of diameter 2 graphs

(1)

The decay number and the maximum genus of

diameter 2 graphs

Hung-Lin Fu

a;∗;1

_{, Ming-Chun Tsai}

b;1

_{, N.H. Xuong}

c

a_{Department of Applied Mathematics, National Chaio Tung University, 1001 TA Hsueh Road,}

Hsinchu 30050, Taiwan

b_{Department of Business Administration, Chung Hua University, Hsinchu, Taiwan} c_{Laboratoire Leibniz-IMAG, 46 Avenue Felix Viallet, 38031 Grenoble Cedex, France}

Received 16 November 1995; revised 16 March 2000; accepted 10 April 2000

Abstract

Let (G) (resp. (G)) be the minimum number of components (resp. odd size components) of a co-tree of a connected graph G. For every 2-connected graph G of diameter 2, it is known that m(G)¿2n(G) − 5 and (G)6(G)64. These results deÿne three classes of extremal graphs. In this paper, we prove that they are the same, with the exception of loops added to vertices.

c

MSC: Primary 05C10; Secondary 05C70

Keywords: Decay number; Betti deÿciency; Diameter; Extremal graphs

1. Introduction

Throughout this paper, a graph may have multiple edges or loops. It is said to be simple if it contains neither multiple edges nor loops.

Let G be a graph with n(G) = |V (G)| vertices and m(G) = |E(G)| edges. Murty [2] (see also [1]) proved the following result.

Theorem 1.1 (Murty [2]). If G is a 2-connected graph of diameter 2; then m(G)¿2n(G) − 5:

Let A be a subset of E(G) and let G − A denote the spanning subgraph obtained from G by deleting all edges in A. Let c(G − A) be the number of components

∗_{Corresponding author.}

E-mail addresses: [email protected] (H.-L. Fu), [email protected] (M.-C. Tsai), [email protected] (N.H. Xuong).

1_{Research supported by National Science Council of the Republic of China (NSC 84-2121-M009-013).}

(2)

of G − A. The Betti number (or cycle rank) of G − A is deÿned by (G − A) = m(G) − |A| − n(G) + c(G − A).

For G connected, let T be a spanning tree of G. Denote by (G − T) the number of components with an odd number of edges of the co-tree G − T. Let (G) be the minimum value of (G − T) over all co-trees of G. The invariant (G), called the Betti deÿciency of G, was ÿrst introduced in Ref. [10] to calculate the maximum genus M(G) of G by the formula M(G) = ((G) − (G))=2.

Motivated by this result, ÄSkoviera [7] deÿned the decay number of G; (G), to be the minimum value of c(G−T) over all co-trees of G. Clearly, (G)=2n(G)−m(G)− 1 + min{(G − T)}. It follows that

Theorem 1.2 (ÄSkoviera [8]). If G is a connected graph; then (G)¿2n(G)−m(G)−1 and equality holds if and only if G admits an acyclic co-tree.

For 2-connected graph G of diameter 2, ÄSkoviera [8] gave a tight upper bound on (G), and hence (G).

Theorem 1.3 (ÄSkoviera [8]). If G is a 2-connected graph of diameter 2; then (G)6(G)64:

It is interesting to note that the preceding bound, together with Theorem 1.2, yields an another proof of Theorem 1.1.

For general G, NebeskÃy [5] discovered a formula to calculate (G). Theorem 1.4 (Nebesky [5]). For any connected graph G;

1 + (G) = max{2c(G − A) − |A| | A ⊆ E(G)}:

As above, one notes that the preceding formula, together with Theorem 1.3, yields another proof of Theorem 1.1 (take A = E(G) in Theorem 1:4).

This paper concerns the extremal graphs of Theorems 1.1 and 1.3. We will prove that they are the same, with the exception of loops added to vertices.

2. Extremal 2-connected graphs of diameter 2

A 2-connected graph G of diameter 2 is called extremal if and only if m(G) = 2n(G) − 5. By Theorem 1.1, such a graph is simple.

Remark 2.1. Let G be a connected graph with m(G) = 2n(G) − 5. Then the diameter of G is at least 2. Moreover, it has at least 4 vertices and its minimum degree is at most 3.

(3)

Fig. 1.

Theorem 2.2 (Palumbiny [6]). Let G be a simple graph of diameter 2 with minimum degree at least 3. Then m(G) = 2n(G) − 5 if and only if G is the Petersen graph.

Murty [3] characterized extremal 2-connected graphs of diameter 2. His result can be stated as follows.

Theorem 2.3 (Murty [3]). The following statements are equivalent for a graph G. (1) G is an extremal 2-connected graph of diameter 2.

(2) G is either the Petersen graph or is constructed by connecting all vertices of K2

or K3 to a new vertex by paths of length 2.

Examples: In Fig. 1, K2 and K3 are in heavy lines, 1 is the new vertex.

3. Two-connected graphs of diameter 2 and decay number 4

In this section, we proceed to characterize 2-connected graphs G of diameter 2 satisfying the equality (G) = 4.

Before stating this characterization, it will be convenient to introduce the following concept.

Deÿnition 1. Let G be a connected graph. We say that a subset A of E(G) is -minimal if 1 + (G) = 2c(G − A) − |A| and, for every B ⊂ A; 1 + (G) ¿ 2c(G − B) − |B|.

The following remark will prove useful subsequently.

Remark 3.1. Let G be a connected graph and let A be a -minimal subset of E(G). Then each component of G − A is an induced subgraph of G and any two dierent components are joined by at most one edge in A.

We are now prepared to describe the family of 2-connected graphs G of diameter 2 with (G) = 4.

(4)

Fig. 2.

Theorem 3.2. Let G be a 2-connected graph of diameter 2. Then (G)=4 if and only if G is an extremal 2-connected graph of diameter 2 with loops added to vertices. Proof. First let B(G) be the set of loops of G and suppose that G − B(G) is an extremal 2-connected graph of diameter 2. Then the removal of A=E(G)−B(G) from G results in a graph with n(G) components. Since |A| = 2n(G) − 5, Theorem 1:4 gives (G)¿4. By Theorem 1.3, equality must hold and the suciency of the condition is proved.

Conversely, let G be a 2-connected graph of diameter 2 with (G) = 4. Let A be a -minimal subset of E(G). Let {C1; C2; : : : ; Cp} be the set of components of G − A.

Then |A| = 2p − (1 + (G)) = 2p − 5. It follows that the loopless graph obtained from G by contracting each Ci to a single vertex veriÿes Remark 2.1. So p¿4 and there

are two components that are not joined by an edge in A. We are going to show that p = n(G), or equivalently, every Ci has only one vertex. Suppose, on the contrary,

there is a component Ck with |V (Ck)|¿2. Since G is 2-connected, there must be two

disjoint edges joining Ck to the remainder of the graph. Let a and b be their endvertices

in V (Ck). Now contract each component Ci (i 6= k) to a single vertex, then identify

any vertex in V (Ck) − {a} with b (loops are deleted and multiple adjacencies between

a and b are replaced by a single edge). Let H be the resulting graph. Then H has diameter 2 since it is not complete. We now show that H is 2-connected. Assume not. Then H has a cutvertex, say Cp, which is adjacent to any Ci. This implies that

the removal of Cp from G results in a graph with at least 2 components G1 and G2

(Fig. 2).

It follows that there are two disjoint subsets I and J of {1; 2; : : : ; p − 1} such that G1 and G2 are spanned, respectively, by S_i∈ICi and S_j∈JCj. Let, say C1, be any

component contained in G1. By Remark 3.1, let u ∈ V (C1) and v ∈ V (Cp) be the

endvertices of the unique edge between C1 and Cp. Now there is in G2 a component,

let C2, which is not joined to v by an edge; for otherwise, v would be a cutvertex

(5)

vertex of C2 is at distance ¿3 from u, contradicting the fact that G has diameter 2.

In conclusion, H is a 2-connected diameter 2 graph. But this contradicts Theorem 1.1 because m(H) = |A| + 1 = (2p − 5) + 1 = 2(p + 1) − 6 = 2n(H) − 6 and the theorem is proved.

On the other hand, the above result can be also proved by an extension of Theorem 1.1 which was obtained by Tsai [9]. Before stating this method, we need the following deÿnition.

Deÿnition 2. Let G be a connected graph. We say that a subset A of E(G) is E-minimal if any two dierent components of G−A are joined by at most one edge in G. Further-more, we denote by G=A the graph obtained from G by contracting each component of G − A into a vertex.

The following remark is an extension of Theorem 1.1.

Remark 3.3 (Tsai [9]). Let G be a 2-connected graph of diameter 2 and let A be an E-minimal subset of E(G). Then

m(G=A)¿2n(G=A) − 5 + i(G=A);

where i(G=A) is the number of components in G − A containing at least two vertices of G.

Remark 3.4. By Remark 3.1, any -minimal subset of E(G) is also an E-minimal subset of E(G). To prove the necessity of Theorem 3.2, one can let A be an -minimal subset of E(G). Then (G) = 2n(G=A) − 1 − m(G=A) = 4. This implies i(G=A) = 0 and m(G=A) = 2n(G=A) − 5. Hence Theorem 3.2 can be obtained by Remark 3.3.

4. Two-connected graphs of diameter 2 and Betti deÿciency 4

We conclude this paper with a characterization of 2-connected graphs G of diameter 2 satisfying the equality (G) = 4. Notice here a formula discovered by NebeskÃy [4] to calculate (G).

Theorem 4.1 (Nebesky [4]). For any connected graph G; 1+(G)=max{c(G −A)+ o(G −A)−|A) | A ⊆ E(G)}; where o(G −A) denotes the number of components of odd Betti number.

As above, the following fact may prove useful.

Remark 4.2. Let G be a connected graph and let A be a -minimal subset of E(G) (i.e. 1 + (G) = c(G − A) + o(G − A) − |A| and, for every B ⊂ A; 1 + (G) ¿

(6)

Fig. 3.

c(G − B) + o(G − B) − |B|). Then each component of G − A is an induced subgraph of odd Betti number (i.e. c(G − A) = o(G − A)) and any two dierent components are joined by at most one edge in A.

Now, we state

Theorem 4.3. Let G be a 2-connected graph of diameter 2. Then (G) = 4 if and only if G is an extremal 2-connected graph of diameter 2 at each vertex of which an odd number of loops are added.

Proof. A proof can be readily supplied by imitating that of Theorem 3.2. Here, let us apply Theorems 2.3 and 3.2 to present a short one of the non-trivial part of the statement.

Let G be a 2-connected graph of diameter 2 with (G) = 4. Then (G) = 4 by Theorem 1.3. Hence, we know from Theorem 3.2 that G arises from an extremal 2-connected graph H of diameter 2 by adding loops to vertices. We now show that if some vertex of G has an even number of loops then (G)63. To see this we shall show, in fact, that for any vertex x of an extremal 2-connected graph H of diameter 2, there is a co-tree K with (K) = 4 such that the component containing x has only one vertex. Using Theorem 2.3, this follows immediately from the constructions illustrated in Fig. 3 below.

(7)

(a) a co-tree of the Petersen graph.

(b), (c) co-trees of the graph built up from K2 by connecting 2 and 3 to 1 by two

2-paths.

(d), (e), (f) co-trees of the graph built up from K3 by connecting 2 and 4 to 1 by

two 2-paths, 3 to 1 by one 2-path.

Remark 4.4. By Remark 4.2, any -minimal subset of E(G) is also an E-minimal subset of E(G). To prove the necessity of Theorem 3.2, one can let A be a -minimal subset of E(G). Hence Theorem 4.3 can also be obtained by Remark 3.3 as Remark 3.4.

Acknowledgements

The authors are grateful to the referees for improving the paper. References

[1] F. Gliviak, A new proof of one estimation, Istituto Lombardo (Rend. Sc.) A 110 (1976) 3–5. [2] U.S.R. Murty, On some extremal graphs, Acta Math. Acad. Sci. Hung. 19 (1969) 69–74.

[3] U.S.R. Murty, Extremal nonseparable graphs of diameter 2, in: F. Harary (Ed.), Proof Techniques in Graph Theory, Academic Press, New York, 1969, pp. 111–118.

[4] L. NebeskÃy, A new characherization of the maximum genus of a graph, Czech. Math. J. 106 (1981) 604–613.

[5] L. NebeskÃy, A characterization of the decay number of a connected graph, Math. Solvaca 45 (4) (1995) 349–352.

[6] D. Palumbiny, Sul numero minimo degli sigoli di un singramma di ragio ediametro equali a due, Istituto Lombardo (Rend. Sci.) A 106 (1972) 704–714.

[7] M. ÄSkoviera, The maximum genus of graphs of diameter 2, Discrete Math. 87 (1991) 175–180. [8] M. ÄSkoviera, The decay number and the maximum genus of a graph, Math. Slovaca 42 (4) (1992)

391–406.

[9] M.C. Tsai, A study of maximum genus via diameter, Ph.D. Thesis, National Chiao Tung University, Taiwan, 1996.

[10] N.H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory Ser. B 26 (1979) 217–225.