Some New Classes of Open Distance-Pattern Uniform Graphs

M (V) for all u, V ∈ V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpuchordal 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 G can be embedded into a self-complementary odpu-graph H, such that G and G are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.


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 distancepattern 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    () = {(, V) : V ∈ ,  ̸ = V}, 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 distancepattern uniform (odpu-) set.The minimum cardinality of an odpu-set in , if it exists, is the odpu-number of  and is denoted by od().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 V in a connected graph , the eccentricity (V) of V is the distance to a vertex farthest from V. 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 V in a connected graph  is called a central vertex if (V) = ().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    () = {1, 2, . . ., ()}, 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.
Theorem 5 (see [2]).The shadow graph of any complete graph   ,  ≥ 3 is an odpu-graph with odpu-number  + 2 (the shadow graph () of a graph  is obtained from  by adding for each vertex V of  a new vertex V  , called the shadow vertex of V, and joining V  to all the neighbors of V in ).
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 2 and diameter 2 or 3, (ii)  has diameter 3 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 3 is never greater than the number of vertices of eccentricity 2.
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  4 -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 0 or 1 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.[4][5][6][7]).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 4 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  0 (, , ) 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  0 ∈  0 (, , ), then the following conditions hold: (1)  0 ⊆ (), (2) there are at least two distinct vertices  1 ,  2 ∈  0 such that for every  = 1, 2 either   (,   ) = 1 or   (,   ) = 1.In particular, | 0 | ≥ 2.
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.
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 (V 1 , V 2 , V 3 ) such that there exists a path connecting V  and V  that contains no neighbor of V  ( ̸ =  ̸ = ), for every combination of 1 ≤ , ,  ≤ 3 (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.

Main Results
The following theorem gives a complete characterization for the odpu-self-complementary graphs.Further, given any positive even integer  ̸ = 2, there exists an odpu, selfcomplementary graphs with odpu-number .Also, we prove that it is possible to embed any graph  into a selfcomplementary, 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 𝐺 is an odpu graph if and only if 𝐺 has no universal vertex in
, where  = ().Hence, all the vertices of  must be adjacent to V in .Hence, V 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  =  = 2 or  = 2 and  = 3.If  =  = 2, then the graph  is self-centered, and hence it is an odpu-graph.
If  = 2 and  = 3, then by Proposition 7,  has a dominating edge V.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.
By Theorem 17 and Proposition 7, the following Corollary holds immediately.

Corollary 18. A self-complementary graph 𝐺 is an odpu-graph if and only if 𝑟
Theorem 19.Given any even integer  ≥ 4, there exists a selfcomplementary, odpu-graph with odpu-number .
Proof.First, we take a path  4 and replace the end vertices of the path  4 by copies of  2 and the interior vertices by copies of the complete -partite graph  2,2,...,2 ;  ≥ 2.Where two vertices of  4 were joined by an edge, the corresponding graphs are now joined by all possible edges between them.Let the resulting graph be  1 .Clearly,  1 is a self-complementary graph of 8 vertices with diameter 3.Moreover, if we add a  2 and join it to all the vertices of the copies of  2,2,...,2 , we get a self-complementary graph  2 of order 8 + 2 and diameter 3 (see Figure 2).Now, we claim that the odpu-number of  1 is 4 and that of  2 is 4+2.Since the eccentricities of the vertices in  2,2,...,2 are 2 and the eccentricities of the vertices of  2 are 3 in  1 , the center ( 1 ) is the collection of all vertices of both 2,2,...,2 .Since ⟨( 1 )⟩ does not have a universal vertex, by Theorem 17,  1 is an odpu-graph.Let  be a minimal odpuset of  1 .For any vertex  ∈ ( 1 ), there is exactly one vertex V ∈ ( 1 ) such that (, V) = 2. Hence, all the vertices of (G 1 ) must be in .Hence, od( 1 ) = 4.Now, consider  2 .By the same argument above, all the vertices of ( 2 ) must be in the minimal odpu-set .Hence, the odpu-number of  2 is 4 + 2, hence, the theorem.
Theorem 20.Any graph  with () ≥ 2 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  4 , and replace the end vertices of the path  4 by copies of  and the interior vertices by copies of .Whenever two vertices of  4 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 selfcomplementary with  (and ) as induced subgraph of  with () = 2 and () = 3.Also, all the vertices of  belong to the centre (), and the vertices of  belong to  − ().Let  = ().
Remark 21.Suppose that the given graph is of radius 1.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 1, 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 2, 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 selfcomplementary, odpu-graph  with  and  being induced subgraphs of .
The following result gives the characterization for odpucographs and proves that the odpu-number of an odpucograph 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, () = 1.Since  is an odpu-cograph, |()| ≥ 2, 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 od() = 2.
So assume that  does not have a universal vertex.Thus, there is no leaf attached to the root (1), and hence, () = 2. 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,  1 ,  2 , . . .,   , which are labeled by (0).Let  be the minimal odpu-set of .Since () = 2,    () = {1, 2} 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 (, ) = 2 only when  and  are descending from same   .So, there exist at least two vertices  and  descending from each   (1 ≤  ≤ ) and belonging to the minimal odpu-set .Hence, || ≥ 2 .Now, we prove that  has exactly 2 elements.Let  be the set of 2 leaves such that the exactly two leaves  and  are descending from the same node   , (1 ≤  ≤ ) with  ∉ ().Let  ∈ ().Then, there exists a  ∈  which is descending from the same node   , where the leaf  belongs, such that  ∉ ().Hence, (, ) = 2, and hence, 2 ∈    ().Since  is adjacent to all vertices in  which are descending from the nodes   , for all  ̸ = , 1 ∈    ().Hence,    () = {1, 2} for all  ∈ ().Hence, od() = 2.
Next, we establish the characterization for odpu-chordal graphs.
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.
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.
Remark 33.The converse of Corollary 32 need not be true.For example,  4 is distance-hereditary and ⟨( 4 )⟩ =  2 , which is self-centered, but  4 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.
Proof.By Proposition 16, the central subgraph ⟨()⟩ of a maximal outerplanar graph  is isomorphic to one of the seven graphs given in Figure 1.

Theorem 35. A maximal outerplanar graph 𝐺 is an odpugraph if and only if it is isomorphic to one of the graphs in
Case 1 (() = (⟨()⟩) = 1).By Proposition 1, there exist at least two universal vertices in , and hence, ⟨()⟩ is isomorphic to either  1 or  2 in Figure 3. Consider all graphs  with two universal vertices.Then,  2 is the least of them.Let  2 ≅  1 be an edge V.Now, add vertices one by one to get a new maximal outerplanar graph  in such a way that  and V are universal in .That is,  = V + {V 1 , V 2 , . . ., V  }, where "+" denotes the operator join (The join of two graphs  1 = ( 1 ,  1 ) and  2 = ( 2 ,  2 ) denoted by  1 +  2 has the vertex set as  =  1 ∪  2 , and the edge set  contains all the edges of  1 and  2 together with all edges joining the vertices of  1 with the vertices of  2 .) of two graphs.When  = 1, which is also a maximal outerplanar-odpu-graph.When  ≥ 3,  = V + {V 1 , V 2 , . . ., } ≅  2 +   ,  ≥ 3 are not maximal outerplanar graphs.Thus,  1 ,  2 , and  3 are the only maximal outerplanar graphs with () = (⟨()⟩) = 1.
Theorem 36.For a maximal outerplanar graph , the odpunumber is either 2 or 5.
Proof.By Theorem 25, the maximal outerplanar odpu-graphs are one of the graphs in Figure 4.

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,  4 -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 2 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.