In this paper, the concept of pathos lict graph of a tree is introduced. We present a characterization of those graphs whose pathos lict graphs are planar, outerplanar, maximal outerplanar, crossing number one, eulerian and hamiltonian

Chandrasekhar, R.

Muddebihal, M.H.

ePrints@Bangalore University

On Pathos Litact Graph of a TreeVenkanagouda M. Goudar *Department of MathematicsSri Siddhartha Institute of TechnologySri Gouthama Research Center, TumkurKarnatakaIndiaAbstractIn this paper, the concept of pathos litact graph of a tree is introduced. We present a characterization of those graphs whose pathos litact graphs are planar, outerplanar, maximal nonouterplanar, crossing number, eulerian and hamiltonian.Keywords: Line graph/lict graph/litact graph/pathos lict graph/pathos litact graph/outerplanar/cross-ing number.Mathematics Subject Classification: 05C10, 30.1. IntroductionThe concept of pathos of a graph G was introduced by Harary [1] as a collection of minimum number of edge disjoint open paths whose union is G. The path number of a graph G is the number of paths in the pathos. The path number of a tree T is equal to k where 2k is the number of odd degree vertices of T. Also the end vertices of each path of any pathos of a tree are of odd degree.  The edge degree of an edge {u, v} of a tree T is the sum of the degrees of u and v. The pathoslength is the number of edges which lie on a particu-lar path Pi of pathos of T. A pendent pathos is a path Pi of pathos having unit length which corresponds to a pendent edge in T. A pathosvertex is a vertex in Pm(T) corresponding to the boundary of the exterior region in any embedding of G in the plane. A graph is said to be minimally non-outerplanar if i(G) = 1. The lict graph [4] n(G) of a graph G is the graph whose vertex set is the union of the set of edges and the set of cutvertices of G in which two *E-mail:  vmgouda@gmail.com©Journal of Discrete Mathematical Sciences & CryptographyVol. 18 (2015), No. 1 & 2, pp. 47–53DOI : 10.1080/09720529.2013.87882148 V. M. GOUDARvertices are adjacent if and only if the corresponding edges of G are adja-cent or the corresponding members of G are incident.The pathos line graph of a tree T denoted by PL(T) is defined as the graph whose vertex set is the union of the set of edges and paths of pathos of T in which two vertices are adjacent if and only if the corresponding edges of T are adjacent and the edge lies on the corresponding path Pi of pathos. The pathos lict graph [5] of a tree T denoted by Pn(T) is defined as the graph whose vertex set is the union of the set of edges, set of paths of pathos and set of cutvertices of T in which two vertices are adjacent if and only if the corresponding edges of T are adjacent and the edge lies on the path of pathos and the edges are incident to the cutvertex. The litact graph [4] m (T) of a graph G is the graph whose vertex set is the union of edges and the set of cut vertices of G in which two vertices are adjacent if and only if the corresponding edges of G are adjacent or the corresponding members of G are incident or the cutvertices are adjacent. The pathos litact graph of a tree T denoted by Pm(T) is defined as the graph whose vertex set is the union of the set of edges, set of paths of pathos and set of cutvertices of T, in which two vertices are adjacent if and only if the corresponding edges of T are adjacent, the edge lies on the path of pathos, edges are incident to the cutvertices and the cutvertices are adjacent. Since the system of path of pathos for a tree T is not unique, the corresponding pathos litact graph is also not unique. The pathos litact graph is defined for a tree having at least two cutvertices All the undefined terms may be referred to Harary [2]. All graphs considered here are finite, undirected and without loops or multiple edges. 2. Preliminary resultsWe need the following Theorems for the proof of our further results. Theorem 2.1 [Ref.3]. If G is a (p. q) graph whose vertices have degree di then its pathos line graph PL(T) has (q + k) vertices & ½ ∑di2 edges, where k is the path number. Theorem 2.2 [Ref.3]. The pathos line graph PL (T) is planar if and only if Δ (T)<4 for every vertex v of T.PATHOS LITACT GRAPH OF A TREE 49Theorem 2.3 [Ref.4]. For any graph G, the litact graph m (G) is minimally out-erplanar if and only if G is a path. Theorem 2.4 [Ref.4]. The litact graph m (G) of a graph G is planar if and only if G is planar and the degree of each vertex is at most 3. Theorem 2.5 [Ref.4]. For any graph G, the litact graph m (G) is maximal outer-planar if and only is G is a path.3. Pathos litact graphsRemark 3.1. For any tree T with p ≥ 3 vertices L(T) ⊂ PL(T) ⊂ Pm(T) and L(T) ⊂ n(T) ⊂ m(T) ⊂ Pn(T) ⊂ Pm(T). Remark 3.2. If the edge degree of an edge uv in a tree T is even (or odd) and u and v are cutvertices then the corresponding vertex in Pm (T) is of odd (or even) degree. Remark 3.3. If the edge degree of an edge in a tree T is even (or odd) and the edge is a pendent edge then the corresponding vertex in Pm(T) is of even (or odd) degree. Remark 3.4. The degree of pathosvertex in Pm(T) is equal to the pathoslen-gth of the corresponding path pi of pathos of T. Remark 3.5. Every pendent pathos in a tree T adds one cutvertex to Pm(T) and the number of cutvertices in Pm(T) is the number of pendent pathos in T. In the following Theorem we obtain the number of vertices and edges in a pathos litact graph.Theorem 3.6. For any (p, q) graph T whose vertices have degree di and cutvertices c have degree cj , the pathos litact graph Pm (T) has (q+k+c)vertices and 12∑di 2 +∑cj +(c-1) edges, where k is the path number.Proof. By Theorem 1, PL(T) has (q + k) vertices and 12∑di2 edges. The number of vertices of Pm(T) is the sum of number of vertices of PL(T) and the number of cutvertices c in T. Hence Pm(T) has (q + k + c) vertices.The number of edges in Pm(T) is the sum of edges in PL(T), the num-ber of edges incident with cutvertices in T and the number of edges in c(G). By Theorem 2.1, the number of edges in PL(T) is ∑di2, the number of edges incident to cutvertices is ∑cj and the c(G) has ∑ c cj j( )−12 edges. 50 V. M. GOUDARSince T contains at most two adjacent vertices, ∑c cj j( )−12+(c – 1). Hence Pm(T) has 12∑di2 +∑cj + (c – 1) edges.The following Lemma gives the number of regions in pathos litact graph. Lemma 3.7. For any (p. q) graph T, the number of regions in pathos litact graph Pm(T) is (p + q + c – 3). In the following theorems we obtain the condition for planarity of pathos litact graph.Theorem 3.8. The pathos litact graph pm(T) of a tree T is planar if and only if Δ (T) < 3, for every vertex v of T.Proof. Suppose Pm(T) is planar. Assume Δ (T) < 4. If there exists a vertex v of degree 4, then by Theorem 2.2, PL(T) is planar and contains K4 as in-duced subgraph. In Pm(T), the vertex v having degree 4 is adjacent to all the vertices of 〈K4〉 and it gives 〈K5〉 as subgraph. Clearly Pm(T) is nonpla-nar, a contradiction. Conversely, suppose Δ (T) < 3. By Theorem 2.4, m(T) is planar. Each block of m(T) is either K3 or K4. The pathosvertex is adjacent to almost two vertices of each block of m(T). This gives a planar Pm(T). We now present a characterization of trees whose pathos litact graphs are non outer planar and minimally nonouterplanar. Theorem 3.9. For any tree T, the pathos litact graph Pm (T) is nonouterplanar.Proof. Suppose Pm (T) of a tree T is outerplanar. We use contradiction for the proof of the theorem. Now we consider the following cases:Case 1. If T is a path Pt of length t ≥ 3. By Theorem 2.5, m (T) is maxi-mal outer planar and the pathosvertex is adjacent to (t-1) vertices. Hence Pm (T) has inner vertices, which is nonouterplanar.Case 2. If Δ(T) ≤ 3 , for any vertex v of T. By Theorem 2.4, m (T) is planar. The vertex v having degree 3 is adjacent to all the vertices to form 〈K3〉 as a subgraph. Also the pathosvertex is adjacent to all vertices of 〈K3〉 to form 〈K4〉 as a subgraph, which is nonouterplanar, a contradiction. Hence Pm (T) is nonouterplanar.PATHOS LITACT GRAPH OF A TREE 51Theorem 3.10. The pathos litact graph Pm (T) of a tree T is minimally nonouter-planar if and only if T is a path of length 3.Proof. Suppose Pm (T) is minimally nonouterplanar. Assume that T is a path Pt for t >3. By Theorem 3.8, Pm (T) has at least (t–2) inner vertices, i.e. i [Pm(T)] >1, a contradiction. Thus t = 3.Conversely, suppose T is a path of length 3. Then Pm(T) has (3-2) =1, inner vertex. Hence i [Pm(T)] = 1, which is minimally nonouterplanar.The noneulerian property of Pm (T) is given by the following Theorem.Theorem 3.11. For any tree T, the pathos litact graph Pm (T) is noneulerian.Proof. Suppose T is a tree with p<3 vertices, then Pm (T) = PL (T) = K1 or K2 and the result is obvious.Suppose T is a tree with p = 3 vertices, then Pm (T) = K4 – x which is noneulerian. Suppose T is a tree with p ≥ 4 vertices, then there exists at least two cutvertices v1 and v2 such that an edge q = v1v2 between these two vertices. We consider the following cases:Case 1. If edge degree of q is even, then by the Remark 2, the correspond-ing vertex in Pm (T) is of odd degree vertex. Hence Pm (T) is noneulerian.Case 2. If edge degree of q is odd, then it contains one pendent edge. By definition, Pm (T) contains an odd degree vertex. Hence Pm (T) is noneu-lerian.From the above two cases, Pm (T) is noneulerian.In the following Theorem we obtain a condition for Pm (T) to be a block.Theorem 3.12 The pathos litact graph of a tree is a block if and only if every cut-vertex of T is of even degree.Proof. Suppose the pathos litact graph of a tree T is a block. Assume that T has one cutvertex of odd degree ≥ 3. Since every path of pathos of T starts and ends with an odd degree vertex, there is a pendent pathos in T coin-ciding with the odd degree vertex . By the Remark 3.5, Pm (T) contains at least one cutvertex. Thus Pm (T) is not a block, a contradiction.Conversely, suppose T has all its cutvertices of even degree 2n. Then n number of paths of pathos passes through these cutvertices and the pa-thoslength of every path is at least two. Since there is no pendent pathos 52 V. M. GOUDARin T and none of the non end edges coincide with an end edge of path of pathos. By the Remark 3.5, Pm (T) has no cutvertex. Thus Pm (T) is a block.The next Theorem Characterizes Pm (T) in terms of crossing number.Theorem 3.13: The pathos litact graph of a tree T has crossing number two if and only if Δ (T) ≤ 4 for every vertex v of T and has a unique vertex of degree 4. Proof. Suppose Pm (T) has crossing number two. We have the following cases:Case 1. Assume that T has vertex of degree 5. Then the edges incident to this vertex together with the cut vertex form 〈K6〉 as subgraph in m (T). The pathos vertex is adjacent to at least two vertices of 〈K6〉 in Pm (T). Clearly C [Pm (T)] > 2, a contradiction.Case 2. Assume that T contains two vertices v1 and v2 of degree 4. Then T has two K1,4 as subgraphs. Then the cutvertices v1 and v2 together with their corresponding incident edges form two subgraphs as K5 in Pm (T). Cleary C [Pm(T)] > 2, a contradiction. Thus T has exactly one cutvertex of degree 4.Conversely, suppose T holds both the conditions of the Theorem. Then by necessity the result follows. In the next Theorem, we characterize hamiltonian Pm (T). Theorem 3.14. The pathos litact graph Pm (T) of a tree T is hamiltonian if and only if every pathos of path length is at least 2. Proof. Suppose Pm (T) is hamiltonian. Assume that T has at least one path such that its pathoslength is one. By the remark 3.5, Pm (T) contains a cut-vertex, a contradiction. Hence, every pathos of path length is at least 2. Conversely, suppose every pathos of path length at least 2. We now consider the following cases: Case 1. If T is a path then it has exactly one path of pathos. Let V [m (T)] = {e1, e2 …..en} È {c1, c2 …..cn-2} where c1, 2…cn-2 are the cutvertices of T. Since each block is a triangle and each block consist vertices as B1= {e1, c1, e2}, B2 = {e2, c2, e3} …. Bm = {en-1, cn-2, en}. In Pm (T) the pathos the pathosvertex is adjacent to [e1, e2 ..en ]. Hence V [Pm (T)] = {e1, e2 ..en} È {c1, c2 .. cn-2} È w form a closed path w e1c1e2c2 . . . . en-1 cn-2 en w containing all the vertices of Pm (T), clearly Pm (T) is hamiltonian. PATHOS LITACT GRAPH OF A TREE 53Case 2. If T is not a path, and every pathos of path Length is at least two, then by the Remark 3.4, degree of every pathos vertex is at least two. Let V [m (T)] = {e1, e2 ..en} È {c1, c2 ..ck} which are cutvertices of T. since T has pi pathosvertices i ≥ 1, each having path length at least two, then V [Pm (T)] contains {e1, e2 ..en} È {c1, c2 ..ck} È {p1, p2 ..pi}, vertices. But each pi is adja-cent to at least two vertices of en and each ci are adjacent to form a cycle containing all the vertices p1, e1, e2, e3, c1, c2, e4….pi of Pm (T). Hence, Pm (T) is hamiltonian. References [1] Harary, F. (1977), Annals of New York Academy of Science,175 :198.  [2] Harary, F (1969), Graph theory, Addison- Wesley Reading, mass.  [3]  Muddebihal, M.H, Gudagudi, B.R and Chandrasekhar, R., (2001), National Academy of Science Letters, 24:116.  [4]  Muddebihal, M.H. and Chandrasekhar. R. (2003), National Academy of Science Letters, Vol. 26, No. 9 and 10. 263, 270.   [5]  V.R.Kulli and M.H. Muddebihal.(2006). Lict Graph and Litact Graph of a Graph, Journal of Analysis and Computation, Vol. 2.No. 133-43.  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On pathos lict graph of a tree

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