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International Journal of Combinatorics
Volume 2013 (2013), Article ID 863439, 7 pages
http://dx.doi.org/10.1155/2013/863439
Research Article

Some New Classes of Open Distance-Pattern Uniform Graphs

Department of Mathematics, S.D. College Alappuzha, University of Kerala, Kerala 688003, India

Received 8 April 2013; Accepted 15 June 2013

Academic Editor: Toufik Mansour

Copyright © 2013 Bibin K. Jose. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given an arbitrary nonempty subset of vertices in a graph , each vertex in is associated with the set and called its open -distance-pattern. The graph is called open distance-pattern uniform (odpu-) graph if there exists a subset of such that for all and is called an open distance-pattern uniform (odpu-) set of The minimum cardinality of an odpu-set in , if it exists, is called the odpu-number of and is denoted by . Given some property , we establish characterization of odpu-graph with property . In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph can be embedded into a self-complementary odpu-graph , such that and are induced subgraphs of . We also prove that the odpu-number of a maximal outerplanar graph is either or .

1. Introduction

All graphs considered in this paper are finite, simple, undirected, and connected. For graph theoretic terminology, we refer to Harary [1].

The concept of open distance-pattern and open distance-pattern uniform graphs was studied in [2]. Given an arbitrary nonempty subset of vertices in a graph , the open -distance-pattern of a vertex in is defined to be the set , where denotes the distance between the vertices and in . If there exists a nonempty set such that is independent of the choice of , then is called open distance-pattern uniform (odpu-) graph, and the set is called an open distance-pattern uniform (odpu-) set. The minimum cardinality of an odpu-set in , if it exists, is the odpu-number of and is denoted by . In this paper, we characterize several classes of odpu graph such as odpu-chordal graphs, interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, ptolemaic graphs, self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We need the following definitions and previous results.

For a vertex in a connected graph , the eccentricity of is the distance to a vertex farthest from . The minimum eccentricity among the vertices of a connected graph is the radius of , denoted by , and the maximum eccentricity is its diameter, . A vertex in a connected graph is called a central vertex if . The collection of all central vertices is called the center of denoted by .

In paper [2], it is proved that a graph with radius is an odpu graph if and only if the open distance-pattern of any vertex in is , and a graph is an odpu-graph if and only if its centre is an odpu-set, thereby characterizing odpu-graphs, which in fact suggests an easy method to check the existence of an odpu-set for a given graph. The central subgraph of a graph is the subgraph induced by the center.

Proposition 1 (see [2]). For any graph , if and only if there exist at least two vertices such that , where denote the degree of the vertex in .

Proposition 2 (see [2]). There is no graph having odpu-number three.

Proposition 3 (see [2]). A graph is an odpu graph if and only if its centre is an odpu set, and hence .

Proposition 4 (see [2]). All self-centered graphs are odpu graphs.

Theorem 5 (see [2]). The shadow graph of any complete graph , is an odpu-graph with odpu-number (the shadow graph of a graph is obtained from by adding for each vertex of a new vertex , called the shadow vertex of , and joining to all the neighbors of in ).

Theorem 6 (see [2]). Every odpu-graph satisfies , where and denote the radius and diameter of , respectively.

The complement of a graph has the same vertices as , and every pair of vertices are joined by an edge in if and only if they are not joined in . A self-complementary graph is one that is isomorphic to its complement .

Proposition 7 (see [3]). Let be a nontrivial self-complementary graph. Then (i)   has radius   and diameter     or   , (ii)   has diameter     if and only if it contains a dominating edge (an edge     is said to be a dominating edge if every edge in     is adjacent to   ), (iii)the number of vertices of eccentricity     is never greater than the number of vertices of eccentricity   .

Cographs (or complement reducible graphs) are defined as the class of graphs formed from a single vertex under the closure of the operations of union and complement. Cographs have the following two remarkable properties. Firstly, they are exactly the -restricted graphs, and secondly, a cograph has a unique tree representation, called a cotree. In the cotree representation, leaves of the cotree represent vertices of the graph. Internal nodes of the cotree are labelled using or in such a way that (0) nodes and (1) nodes alternate along every path starting from the root which is always a (1) node. Each nonleaf vertex of has at least two children. The root will have only one (0) node child if and only if the represented cograph is disconnected. Also, the cotree for a particular cograph is unique up to a permutation of the children of the internal nodes. Two vertices and of the graph are adjacent if and only if the unique path from to the root of the tree meets the unique path from to the root at a (1) node. In order to establish various properties about cographs, we label each internal node of a cotree as follows: the root is labelled (1), the children of a node with label (1) are labelled (0), and children of a node labelled (0) are labelled (1). Henceforth, we assume all cotrees to be labelled as such, and we will refer to the internal nodes of cotrees as (0)-nodes and (1)-nodes (cf. [47]).

Theorem 8 (see [4]). A graph has a cotree if and only if is a cograph. In that case, the cotree representation is unique.

A graph is said to be chordal if every cycle of length at least has a chord, that is, an edge joining nonconsecutive vertices of the cycle. A subset is called a vertex separator of for nonadjacent vertices and (or separator of ) if, in (the graph obtained from by the removal of the vertices of and incident edges), the vertices and belong to distinct connected components. Let be the set of all separators of . If no proper subset of belongs to , then is called a minimal separator of . Let be the set of all minimal separators of . The following propositions (cf. [8]) are needed in proving our main results.

Proposition 9 (see [8]). If is a chordal graph and , are distinct nonadjacent vertices in , the central graph of , and , then the following conditions hold:(1) ,(2)there are at least two distinct vertices     such that for every     either     or   . In particular, .

Proposition 10 (see [8]). If is a chordal graph and , are distinct nonadjacent vertices in , the central subgraph of , then .

Proposition 11 (see [8]). If is a chordal graph, then is connected.

Proposition 12 (see [8]). If is a chordal graph, then .

Proposition 13 (see [8]). If is a chordal graph with , then .

Proposition 14 (see [8]). There are no self-centered chordal graph with . Consequently, for a self-centered chordal graph , .

A distance-hereditary graph (cf. [9]) is a connected graph, which preserves the distance function for induced subgraphs. That is, the distance between any two nonadjacent vertices of any connected induced subgraph of such a graph is the same as the distance between these two vertices in the original graph.

Proposition 15 (see [9]). Let be a distance-hereditary graph, and let be a vertex of . Then, for each positive integer and vertices , of the same connected component of , one has , where the ith neighborhood of the vertex , .

A graph is an interval graph if and only if there is a one-to-one correspondence between its vertices and a set of intervals on the real line, such that two vertices are adjacent if and only if the corresponding intervals have an intersection. It is also well known that a graph is an interval graph if and only if is chordal and asteroidal triple-free, where asteroidal triple is a set of three distinct vertices such that there exists a path connecting and that contains no neighbor of , for every combination of (cf. [10]).

A graph is a split graph if and only if its vertices can be partitioned into an independent set, and vertices which induces a clique. For simplicity, given a split graph , we call a vertex partition of , such that is an independent set and vertices in induce a clique, as an - -decomposition of . It is known that a chordal graph whose complement is also a chordal graph is equivalent to a split graph (cf. [10]).

A graph is strongly chordal if and only if is chordal, and every even cycle of length six or more contains a chord splitting the cycle into two odd length paths (cf. [10]).

Ptolemaic graphs are exactly those graphs that are both chordal and distance-hereditary (cf. [11]).

A graph is outerplanar if it can be drawn in the plane with all nodes in the exterior boundary. It is called maximal outerplanar if no edge can be added without destroying its outerplanar property. Every maximal outerplanar graphs is chordal (cf. [12]).

Proposition 16 (see [12]). If is a maximal outerplanar graph, then its central subgraph is isomorphic to one of the seven graphs in Figure 1.

863439.fig.001
Figure 1

2. Main Results

The following theorem gives a complete characterization for the odpu-self-complementary graphs. Further, given any positive even integer , there exists an odpu, self-complementary graphs with odpu-number . Also, we prove that it is possible to embed any graph into a self-complementary, odpu-graph with and being induced subgraphs of the graph . Recall that a universal vertex means a vertex which is adjacent to all other vertices of the graph.

Theorem 17. A self-complementary graph is an odpu graph if and only if has no universal vertex in .

Proof. Assume that is a self-complementary, odpu-graph. If is a universal vertex in , then , where . Hence, all the vertices of must be adjacent to in . Hence, is an isolated vertex in , a contradiction to the hypothesis that is a self-complementary graph.
Conversely, let the self-complementary graph be without universal vertex in . By Proposition 7, the possible radius and diameter of are or and . If , then the graph is self-centered, and hence it is an odpu-graph.
If and , then by Proposition 7, has a dominating edge . In this case, the end points of the dominating edge lie in the centre, and any vertex outside the centre is adjacent to exactly one end vertex of any dominating edge.
Let be any arbitrary vertex in . By hypothesis, is not a universal vertex, and hence, there exists a vertex such that . Therefore, . Also since is adjacent to a dominating edge, . Hence, for all .
Now, let be an arbitrary vertex in . Then, is adjacent to exactly one vertex of a dominating edge . Hence, either and or and . In both cases, for all , and hence is an odpu-graph.

By Theorem 17 and Proposition 7, the following Corollary holds immediately.

Corollary 18. A self-complementary graph is an odpu-graph if and only if .

Theorem 19. Given any even integer , there exists a self-complementary, odpu-graph with odpu-number .

Proof. First, we take a path and replace the end vertices of the path by copies of and the interior vertices by copies of the complete -partite graph . Where two vertices of were joined by an edge, the corresponding graphs are now joined by all possible edges between them. Let the resulting graph be . Clearly, is a self-complementary graph of vertices with diameter . Moreover, if we add a and join it to all the vertices of the copies of , we get a self-complementary graph of order and diameter (see Figure 2).
Now, we claim that the odpu-number of is and that of is . Since the eccentricities of the vertices in are and the eccentricities of the vertices of are in , the center is the collection of all vertices of both . Since does not have a universal vertex, by Theorem 17, is an odpu-graph. Let be a minimal odpu-set of . For any vertex , there is exactly one vertex such that . Hence, all the vertices of must be in . Hence, .
Now, consider . By the same argument above, all the vertices of must be in the minimal odpu-set . Hence, the odpu-number of is , hence, the theorem.

863439.fig.002
Figure 2

Theorem 20. Any graph with can be embedded into a self-complementary, odpu-graph with both and as induced subgraphs of .

Proof. We construct the graph as follows. First, consider a path , and replace the end vertices of the path by copies of and the interior vertices by copies of . Whenever two vertices of were joined by an edge, the corresponding graphs are joined by all possible edges between them so that we get a complete bipartite graph between the vertices of the corresponding graphs. Then, the resulting graph is self-complementary with (and ) as induced subgraph of with and . Also, all the vertices of belong to the centre , and the vertices of belong to . Let .
Let . Since , there exists a vertex not adjacent to in the same copy of . But there exists a path in , where is in , for which do not belong. Hence, , and therefore, . Since for and , . Hence, for all .
Let . Then, is adjacent to a vertex . Therefore, . Also by construction, there exists a vertex such that . Hence, . Hence, for all . Hence, for all . Hence, is an odpu-graph.

Remark 21. Suppose that the given graph is of radius . Then, the construction given in Theorem 20 gives that has a universal vertex. Hence, by Theorem 17, the resulting graph is not an odpu-graph. So, if the given graph is of radius , then first embed the graph into a graph of radius greater than or equal two, by adding a path in any one of the vertex or by any other methods in such a way that the given graph is an induced subgraph of . Then, the graph is of radius greater than or equal to , and hence, we can apply the above construction given in Theorem 20. Hence, we have the generalized form of Theorem 20, as follows.

Theorem 22. Any graph can be embedded into a self-complementary, odpu-graph with and being induced subgraphs of .

The following result gives the characterization for odpu-cographs and proves that the odpu-number of an odpu-cograph is always even.

Theorem 23. A cograph ( -free graph) is an odpu-graph if and only if .

Proof. Let be a -free odpu-graph. Then, clearly .
Conversely, let be a cograph with . Since the diameter of a cograph is less than or equal to , the following are the only three possibilities. If (i) or (ii) , then is self-centered, and hence, it is an odpu-graph. If (iii) and , then by assumption , there exist at least two universal vertices and in . Therefore, forms an odpu-set of , hence the theorem.

Corollary 24. A cograph is an odpu-graph if and only if .

Theorem 25. The odpu-number of an odpu-cograph is always even.

Proof. By Theorem 8, each cograph is uniquely represented as a cotree and conversely. Hence, we prove the theorem using cotree characterization of cographs. Consider the cotree of the odpu-cograph . Since the odpu-graphs are connected, the root which is labeled by (1) in the cotree has at least two children.
If the root (1) has a child which is a leaf in the cotree, then this leaf vertex is adjacent to all vertices of the cograph . Hence, is a universal vertex in , and hence, . Since is an odpu-cograph, , and hence, there exist two universal vertices in . Hence, there exist at least two leaves attached to the root (1) of the cotree . Hence, in this case .
So assume that does not have a universal vertex. Thus, there is no leaf attached to the root (1), and hence, . Since is connected and all children of the root (1) are labeled by (0), the root (1) has at least two children which are labeled by (0). Let the root (1) have children, namely, , which are labeled by (0). Let be the minimal odpu-set of . Since , for all . Also, for a cotree, each node labeled by (0) at least has two leaves and descending from that node which are nonadjacent. Also, each leaf descending from the node is adjacent to all the leaves descending from , , hence, leaves and with only when and are descending from same . So, there exist at least two vertices and descending from each and belonging to the minimal odpu-set . Hence, .

Now, we prove that has exactly elements. Let be the set of leaves such that the exactly two leaves and are descending from the same node , with . Let . Then, there exists a which is descending from the same node , where the leaf belongs, such that . Hence, , and hence, . Since is adjacent to all vertices in which are descending from the nodes , for all , . Hence, for all . Hence, .

Remark 26. Given any positive even integer , the complete -partite graphs are odpu-cographs with odpu-number .

Next, we establish the characterization for odpu-chordal graphs.

Theorem 27. A chordal graph is an odpu-graph if and only if .

Proof. Let be a chordal graph with = . Since , there are two possibilities:(i) . Then, there are two universal vertices in , and hence by Proposition 1, is an odpu-graph,(ii) . By Proposition 10, for all . Let . Since , there exists a vertex such that . Hence, for all . Now, the connectedness of ensures the existence of a vertex such that . Therefore, for all . Now, let .

Claim 1 ( ). If there is no vertex such that , then is adjacent to all the vertices of . Since , there exist two vertices and in such that . Now, consider . Since , . But contains an ( )-path, which is a contradiction to the fact that is chordal, and hence, . Hence, there exists a vertex such that implies .

Claim 2 ( ). Let be nonadjacent to any of the vertices of . Let such that . Since , . Thus, there exist vertices and in such that and are paths of length in . If , then the path leads to the contradiction. Otherwise the path , a contradiction that . Hence, there exists a vertex such that , and hence, for all . Therefore, for all . Hence, is an odpu-graph.
Conversely, assume that is a chordal-odpu-graph. Since and are connected, .
If , then for all . Thus, there exist at least two universal vertices in . Hence, .
If , for all . Since is an odpu-set, the distance from any vertex of to any vertex of is less than or equal to . Thus, , and hence, .

Corollary 28. A chordal graph is an odpu-graph, and then is self-centered.

Proof. Let be chordal, odpu-graphs. If is not self-centered, then . Hence, there exist vertices such that . Let , , and . Since for all , has an element , a contradiction.

Remark 29. The converse of Corollary 28 need not be true. For example, is chordal with , which is self-centered, but is not an odpu-graph.

Since interval graphs, split graphs, block graphs, ptolemaic graphs, strongly chordal graphs, and maximal outerplanar graphs are subclasses of chordal graphs, (cf. [10]), the following corollary is immediate from Theorem 27 and Corollary 28.

Corollary 30. The following classes of graphs : interval graphs, split graphs, block graphs, ptolemaic graphs, strongly chordal graphs, or maximal outerplanar graphs are odpu-graphs if and only if , and hence, is self-centered.

The following theorem establishes the characterization of odpu-distance-hereditary graphs. Further, we show that the central subgraph of distance-hereditary-odpu-graph is either self-centered or disconnected.

Theorem 31. Let be a distance-hereditary graph with connected . Then, is an odpu-graph if and only if .

Proof. Let be a distance-hereditary graph with   which is connected. Let be an odpu-graph. If , then . Since is connected, for all . But since , for all , which is not possible. Hence, .
Conversely, let . Let . Then, there exists a vertex such that . Since is connected, .
Now, let . If , then , where . Let be the largest integer such that , and let . Then, since , . But since , by Proposition 15, . Thus, , a contradiction. Therefore for all .
Now, if , then . Since , by Proposition 15, for every . Thus, , a contradiction. Hence, .
Since , for all , hence the theorem.

Corollary 32. For a distance-hereditary-odpu-graph , either is disconnected or it is self-centered.

Proof. Let be a distance-hereditary-odpu-graph. If is connected, then we prove that is self-centered. If not, let . Let such that and . Since is distance-hereditary, for all . Thus, and , which is a contradiction, hence the theorem.

Remark 33. The converse of Corollary 32 need not be true. For example, is distance-hereditary and , which is self-centered, but is not an odpu-graph.

The following theorem gives a necessary condition for a maximal outerplanar graph to be an odpu-graph in terms of a specific structure of the central subgraph .

Theorem 34. If is a maximal outerplanar-odpu-graph, then its central subgraph is isomorphic to one of the graphs in Figure 3.

863439.fig.003
Figure 3

Proof. By Proposition 16, the central subgraph of a maximal outerplanar graph is isomorphic to one of the seven graphs given in Figure 1.
Since every maximal outerplanar graph is chordal, by Corollary 28, the central subgraph is self-centered. Therefore, is isomorphic to one of the three graphs given in Figure 3, hence the theorem.

Theorem 35. A maximal outerplanar graph is an odpu-graph if and only if it is isomorphic to one of the graphs in Figure 4.

863439.fig.004
Figure 4

Proof. Let be a maximal outerplanar-odpu-graph. By Theorem 34, is isomorphic to one of the graphs in Figure 3, and by Corollary 30, . Thus, there are two cases: and .

Case 1 ( ). By Proposition 1, there exist at least two universal vertices in , and hence, is isomorphic to either or in Figure 3. Consider all graphs with two universal vertices. Then, is the least of them. Let be an edge . Now, add vertices one by one to get a new maximal outerplanar graph in such a way that and are universal in . That is, , where “+” denotes the operator join (The join of two graphs and denoted by has the vertex set as , and the edge set contains all the edges of and together with all edges joining the vertices of with the vertices of .) of two graphs. When , , which is a maximal outerplanar-odpu-graph. When , , which is also a maximal outerplanar-odpu-graph. When , , are not maximal outerplanar graphs. Thus, , , and are the only maximal outerplanar graphs with .

Case 2 ( ). By Theorem 34, the central subgraph is isomorphic to only. Since is self-centered, it is an odpu-graph. Suppose that there exists a maximal outerplanar-odpu-graph other than with . Then, there exists a vertex such that is adjacent to a vertex of ; say the vertex . Since is maximal outerplanar, cannot be adjacent to any of the vertices , , and . If there exists a vertex such that is a path in the graph , then , a contradiction. Hence, . Thus, , and hence, , a contradiction. Hence, there does not exist a vertex other than the vertices of . Hence, there is no maximal outerplanar-odpu-graph other than with .

Theorem 36. For a maximal outerplanar graph , the odpu-number is either or .

Proof. By Theorem 25, the maximal outerplanar odpu-graphs are one of the graphs in Figure 4.
It is enough to prove that each of , , and has odpu-number . Next, the graph is self-centered with radius and hence, , for all . Let be a minimal odpu-set of . Hence, is the only vertex at a distance of two from the vertex , and hence, must be in . Similarly and are the only vertices of at a distance of two from and , respectively, and hence, . Hence, . If , then , for . Thus, without loss of generality, let . Then, . But , and hence, at least one of the vertices and must be in . So let . Then, . Hence, , for all . Thus, . Hence, the odpu-number of maximal outerplanar graph is either or .

Theorem 37. For every integer , there is an odpu-graph of order such that its central subgraph is disconnected.

Proof. Consider two disjoint complete graphs and of order . Now, add all edges between these two complete graphs, and subdivide each of the new edges of the bipartite subgraph between and by one (a degree vertex) to get a graph of order . Let the vertices of , the vertices of , and the vertex which subdivides the earlier edge of the bipartite graph. Then, all vertices ’s have eccentricity in , and the new graph has radius and diameter . Also the central subgraph is the disjoint union of the complete graphs and . Hence, is disconnected.
Now, we prove that is an odpu-graph. Let . For each , there exists a such that , and hence, . Now, , for all , and hence, . Hence, for all . Similarly, for all . Now, each vertex is adjacent to exactly and , and hence, and . Hence, for all . Since , for all and , for all , for all . Hence, for all , and hence, is an odpu-graph.

3. Conclusion

The characterization of odpu-graphs leads to an interesting condition , for many important classes of graphs such as chordal graphs, interval graphs, split graphs, strongly chordal graphs, self-complementary graphs, -free graphs, maximal outerplanar graphs, ptolemaic graphs, and distance-hereditary graphs. However, this characterization is not in general a characterization for all odpu-graphs. For example, by Theorem 37, there are classes of odpu-graphs with radius and disconnected centre. That is, . Thus, there are more classes of odpu-graphs which do not come under this characterization. We leave it for further scope of investigations.

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