THE MONOPHONIC GRAPHOIDAL COVERING NUMBER OF A GRAPH

Abstract: A chord of a path P is an edge joining two non-adjacent vertices of P . A path P is called a monophonic path if it is a chordless path. A monophonic graphoidal cover of a graph G is a collection ψm of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The minimum cardinality of a monophonic graphoidal cover of G is called the monophonic graphoidal covering number of G and is denoted by ηm. We determine bounds for it and characterize graphs which realize these bounds. Also, for any positive integer n with q − p + 2 ≤ n ≤ q − 1, there exists a tree T such that the monophonic graphoidal covering number is n.


Introduction
By a graph G = (V, E) we mean a finite, undirected connected graph without loops or multiple edges.The order and size of G are denoted by p and q respectively.For basic graph theoretic terminology we refer to Harary [6].The concept of graphoidal cover was introduced by Acharya and Sampathkumar [2] and further studied in [1, 3 ,7 ,8].
A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths in G satisfying the following conditions.
(i) Every path in ψ has at least two vertices.
(ii) Every vertex of G is an internal vertex of at most one path in ψ.
(iii) Every edge of G is in exactly one path in ψ.
The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G and is denoted by η(G).
The collection ψ is called an acyclic graphoidal cover of G if no member of ψ is cycle; it is called a geodesic graphoidal cover if every member of ψ is a shortest path in G.The minimum cardinality of an acyclic (geodesic) graphoidal cover of G is called the acyclic (geodesic) graphoidal covering number of G and is denoted by η a (η g ).The acyclic graphoidal covering number and geodesic graphoidal covering number are studied in [4,5].
A chord of a path P is an edge joining any two non-adjacent vertices of P .A path P is called a monophonic path if it is a chordless path.For any two vertices u and v in a connected graph G, the monophonic distance d m (u, v) from u to v is defined as the length of a longest u−v monophonic path in G.The monophonic eccentricity e m (v The monophonic distance was introduced and studied in [9,10].
The following theorems will be used in the sequal.
Theorem 1.1.[6] Every non-trivial connected graph has at least two vertices which are not cut vertices.Theorem 1.2.[6] Let G be a connected graph with at least three vertices.The following statements are equivalent: (i) G is a block (ii) Every two vertices of G lie on a common cycle.
Throughout this paper G denotes a connected graph with at least two vertices.

Monophonic Graphoidal Cover
Definition 2.1.A monophonic graphoidal cover of a graph G is a collection ψ m of monophonic paths in G such that every vertex of G is an internal vertex of at most one monophonic path in ψ m and every edge of G is in exactly one monophonic path in ψ m .The minimum cardinality of a monophonic graphoidal cover of G is called the monophonic graphoidal covering number of G and is denoted by η m (G).

Example 2.2.
For the graph G given in Figure 2.1, Proof.Since any acyclic graphoidal cover is a graphoidal cover and any monophonic graphoidal cover is an acyclic graphoidal cover, we have η(G) ≤ η a (G) ≤ η m (G).Also, since every geodesic is a monophonic path, we have every geodesic graphoidal cover is a monophonic graphoidal cover and so where n is the number of end vertices of T .All the inequalities in Theorem 2.3 can be strict.For the graph . Now, we proceed to characterize graphs G for which the bounds of η m (G) are attained.
For any monophonic graphoidal cover ψ m of a graph G, let t ψm denote the number of vertices of G which are not internal vertices of any path in ψ m .Let t m = min t ψm , where the minimum is taken over all graphoidal covers of G.
Proof.Let ψ m be any monophonic graphoidal cover of G. Then q = Corollary 2.6.Let T be a tree with n pendant vertices.Then η m (T ) = n − 1.
Corollary 2.7.Let G be a graph having n simplicial vertices.Then η m (G) ≥ q − p + n.Furthermore, equality holds if and only if there exists a monophonic graphoidal cover ψ m of G such that every non-simplicial vertex of G is an internal vertex of a unique monophonic path in ψ m .
The following proposition is the characterization result of the lower bound of η m (G) and it follows from Corollary 2.7.
Proposition 2.8.For any connected graph G of order at least 3, η m (G) = q − p if and only if G has no simplicial vertices and there exists a monophonic graphoidal cover ψ m such that every vertex of G is an internal vertex of a unique monophonic path in ψ m .Theorem 2.9.For any connected graph G, η m (G) = q if and only if G is complete.
Proof.Let G be a complete graph.Since any two vertices of G are adjacent, the length of any monophonic path is one.Hence E(G) is the unique monophonic graphoidal cover of G and so η m (G) = q.
Conversely, suppose that η m (G) = q.Claim that G is complete.If G is not complete, then there exists a monophonic path, say P , in G such that For any connected graph G of order p ≥ 3, Proof.Let η m (G) = q − 1.Since p ≥ 3, by Theorem 1.1 there exists a vertex x, which is not a cut vertex of G.If G has two or more cut vertices, then let P be a monophonic path containing at least two cut vertices.Then |E(P )| ≥ 3. Clearly, ψ m = {E(G) − E(P )} ∪ {P } is a monophonic graphoidal cover of G and so η m (G) ≤ |ψ m | = q − 2, which is a contradiction.Thus the number of cut vertices k of G is at most one.
Case (i): If k = 0, then the graph G is a block.If p = 3, then G = K 3 and so by Theorem 2.9, η m (G) = q, which is a contradiction to the assumption.If p ≥ 4, we claim that G is complete.Suppose that G is not complete.Then there exists two vertices x and y in G such that d(x, y) ≥ 2. By Theorem 1.2, x and y lie on a common cycle and hence x and y lie on a smallest cycle C = x, x 1 , x 2 , ..., y, ..., x n , x of length at least 4. Clearly, all the edges of C lie on either an x − y monophonic path, say P 1 , or an y − x monophonic path, say P 2 .Then ψ m = {E(G) − E(C)} ∪ {P 1 , P 2 } is a monophonic graphoidal cover of G and so η m (G) ≤ q − 2, which is a contradiction.Hence G is complete and so by Theorem 2.9, η m (G) = q, which is again a contradiction.Thus k = 0.
Case (ii): If k = 1, let x be the cut vertex of G.If p = 3, then It is enough to prove that every block of G is complete.Suppose that there exists a block B, which is not complete.Let u and v be two vertices in B such that d(u, v) ≥ 2. Then as in Case (i), η m (G) ≤ q − 2, which is a contradiction.Thus every block of G is complete so that G = K 1 + ∪ m j K j , where K 1 is the vertex x and m j ≥ 2.
Since every monophonic path in K m,n is a geodesic, we have the following result by Theorem 1.3.
Theorem 2.13.Let G be a unicyclic graph with n pendant vertices.Let C be the unique cycle in G having length greater than 3 and let k be the number of vertices of degree greater than 2 on C. Then if there exists two non − adjacent vertices of degree > 2 on C (or) all vertices in C are of degree > 2 n + 1 otherwise.
Proof.Let C : v 0 , v 1 , v 2 , ..., v l , v 0 be the unique cycle in G having length greater than 3.
Case (i) : k = 0. Then G = C and by Theorem 2.11, η m (G) = 2. Case (ii): k = 1.Let v 0 (say) be the unique vertex of degree greater than 2 on C. Let G ′ = G − {v 1 }.Then G ′ is a tree with n + 1 pendant vertices and hence by Corollary 2.6, η m (G ′ ) = n.Let ψ ′ m be a minimum monophonic graphoidal cover of G ′ .Clearly any path in ψ ′ m is a monophonic path in G, we have Also, at least one vertex on C and all the n pendant vertices are exterior vertices of any minimum monophonic graphoidal cover of G, we have t m ≥ n+1.Then by Theorem 2.5, Case (iii): k = 2 and the vertices of degree greater than 2 on C are adjacent in G.
Let v 0 , v 1 be vertices of degree greater than 2 on C.
Clearly G ′ is a tree with n + 1 pendant vertices and hence by Corollary m is a minimum monophonic graphoidal cover of G ′ , then ψ ′ m ∪{P } is a monophonic graphoidal cover of G and hence η m (G) ≤ n+1.Also, at least one vertex on C and all the n pendant vertices are exterior vertices of any minimum monophonic graphoidal cover of G, we have Case (iv): k ≥ 2 and there exists two non-adjacent vertices of degree greater than 2 on C.
Let u, v be vertices of degree greater than 2 on C such that all vertices in a (u − v)section of C other than u, v have degree 2. Let P denote this (u, v)section and let G ′ be the subgraph obtained by deleting all the internal vertices of P. Clearly G ′ is a tree with n pendant vertices and hence by Corollary 2.6, m is a minimum monophonic graphoidal cover of G ′ , then ψ ′ m ∪ {P } is a monophonic graphoidal cover of G and hence η m (G) ≤ n.Also, since G has n pendant vertices, t m ≥ n so that η m (G) = n.
Case (v): k ≥ 3 and all the vertices of C are of degree greater than 2.
Let H = G − {v 1 v 2 , v 2 v 3 }.Let H ′ and H ′′ be the components of H with H ′ contain the vertices v 1 ,v 3 and H ′′ contains the vertex v 2 .Let r be the number of pendant vertices in H ′ and let s be the number of pendant vertices in H ′′ .Since any pendant vertex of H ′ or H ′′ is a pendent vertex of G, we have n Then G ′ contains r pendant vertices and G ′′ contains s + 2 pendant vertices.Clearly G ′ and G ′′ are trees and hence by Corollary 2.6, η m (G ′ ) = r − 1 and η m (G ′′ ) = s + 1.Let ψ ′ m be a minimum monophonic graphoidal cover of G ′ and let ψ m ′′ be a minimum monophonic graphoidal cover of G ′′ .Then ψ ′ m ∪ ψ ′′ m is a monophonic graphoidal cover of G and hence η m (G) ≤ r − 1 + s + 1 = n.Also, since G has n pendant vertices, t m ≥ n so that η m (G) = n.
We have seen that if G is a connected graph of order p ≥ 3, then q − p ≤ η m (G) ≤ q.Also we have η m (G) = q − p if and only if G has no simplicial vertices and there exists a monophonic graphoidal cover ψ m such that every vertex of G is an internal vertex of a unique monophonic path in ψ m and η m (G) = q if and only if G is complete.Also, it is proved that η m (G) = q − 1 if and only if G = K 1 + ∪ m j K j , where m j ≥ 2. In the following theorem, we give an improved bounds for the monophonic graphoidal covering number of a graph interms of its size and monophonic diameter.
For any connected graph G of order p ≥ 2, ⌈q/d m ⌉ ≤ η m (G) ≤ q − d m + 1, where d m is the monophonic diameter of G.
Proof.Let ψ m be a minimum monophonic graphoidal cover of G. Since every edge of G is in exactly one monophonic path in ψ m , we have q = Now we give a realization result for the monophonic graphoidal covering number with some suitable conditions.Theorem 2.15.For any positive integer n with q − p + 2 ≤ n ≤ q − 1, there exists a tree T such that the monophonic graphoidal covering number is n. .Proof.Let P : v 1 , v 2 , v 3 , ..., v q−n+2 be a path of order q − n + 2. Let T be a tree obtained from P by adding n − 1 new vertices u 1 , u 2 , ...u n−1 and joining each vertex u i (1 ≤ i ≤ n−1) to the vertex v q−n+1 .The tree T is given in Figure 2.3 and it has n + 1 pendant vertices.Then by Corollary 2.6, η m (T ) = n.In a tree T, q = p−1 and so q −p, q −p+1 are non-positive numbers.Hence there does not exist a tree T whose monophonic graphoidal covering number is either q − p or q − p + 1.Also, by Theorem 2.9, η m (G) = q if and only if G is complete.Thus there does not exist a tree with the monophonic graphoidal covering number is q − p or q − p + 1, or q − 1.

Figure 2
Figure 2.3: T Remark 2.16.In a tree T, q = p−1 and so q −p, q −p+1 are non-positive numbers.Hence there does not exist a tree T whose monophonic graphoidal covering number is either q − p or q − p + 1.Also, by Theorem 2.9, η m (G) = q if and only if G is complete.Thus there does not exist a tree with the monophonic graphoidal covering number is q − p or q − p + 1, or q − 1.