Characterization and Construction of Permutation Graphs

If  is a permutation of , the graph G 1, 2, ,n    has vertices 1, where 2, , n  xy is an edge of G if and only if  ,  x y or   , y x is an inversion of  . Any graph isomorphic to G is called a permutation graph. In 1967 Gallai characterized permutation graphs in terms of forbidden induced subgraphs. In 1971 Pnueli, Lempel, and Even showed that a graph is a permutation graph if and only if both the graph and its complement have transitive orientations. In 2010 Limouzy characterized permutation graphs in terms of forbidden Seidel minors. In this paper, we characterize permutation graphs in terms of a cohesive order of its vertices. We show that only the caterpillars are permutation graphs among the trees. A simple method of constructing permutation graphs is also presented here.


Introduction
A bijection  of  to itself is called a permutation of order .We shall write    The term graph of inversions was used by Ramos in [1].For our purpose in this paper, any graph isomorphic to G  for some permutation  will be called a permutation graph.There is an implementation Permu-tationGraph[p] in the Combinatorica package of Mathematica [2] that creates the permutation graph p G .In 1967 Gallai [3] characterized permutation graphs in terms of forbidden induced subgraphs.In 1971 Pnueli, Lempel, and Even [4] showed that a graph is a permutation graph if and only if both and its complement G G G have transitive orientations.Recently in 2010 Limouzy [5] gave a characterization of permutation graphs in terms of forbidden Seidel minors.

  
A characterization of permutation graphs in terms of cohesive vertex-set order is established in this paper.We prove that the only permutation graphs among the trees are the caterpillars.In addition, we describe a simple method of constructing permutation graphs.

Cohesive Vertex-Set Order
The vertex-set of a graph will be denoted by G   V G ab while the edge-set will be denoted by .An edge with end-vertices and will be denoted by or .For graph theoretic terms used here without definition, the book by Harary [6] may be referred to.

Consider the permutation
. The inversions of

 
3,1, 4, 6,5, 2 There are some important properties of the drawing that we need to take note of.
(a) If and bc are two edges of the graph where lies between and in the drawing, then is also an edge.S. V. GERVACIO ET AL. 34 (b) If uv is an edge and x is a vertex that lies be- tween u d v in the drawing, then either uv is an edge or an xv is an edge.We de ne more precisely the properties that we obse fi rved.Definition 2.1 Let G be a graph of order n .An ar- rangement der of G (or simply cohesive order G ) if the following conditions are satisfied: graph G  G has the same vertex-set as  2) For all vertices x and y in T , if   Our main result, hich aract izes graphs, is the following theorem Conversely, let G be a graph with a hesive der i o or By p   of a cohesive order, it follows that D to a tournament by orient- , where  is an inversion, we have a b  , or . Consely, the ed e gra in der  1 5 x x x x x .To be able to follow the discussion in the proof of theorem without difficulty, let  dir s om left to right.For two vertices not adjacent in we assign the ar that goes fr x is 3.The number of westbound arcs is simply the number of vertices to its left that are not adjacent to to 4 x .The table below summarizes the scores of the vertices. Vertex Take the pe io

Construction and Examples of Permutation G
Some fundamental facts about permutatio given in the next theorem.Theorem 3.1 Let G be a equivalent: (a) G is a permutation is a cohesive order of , then the su se , , , , where 1 2 , , , , , , , , , , , , , , , is a cohesive order o .There n f fore is a permutation graph.
We can now identify permutation graphs throu e existence of a cohesive order.Moreover, we can even hic to the graph having a cohesive order.In the drawing of the path n P , we have Condition (a) is vacuously satisfied because there is no is a permutation graph.K into two edges, as shown in Figure 5.
It is not difficult to argue indirectly that 1,3 K  has no cohesive order.Therefore this is not permutation graph.This result is also established by mouzy [5] where he used the symbol T fo a Li 2 r 1,3 K  .Harary and Schwenk [9] defined a caterpillar to be a tree with the property that the removal of all pendant vertices results into a path.Figure 6 shows a caterpillar with 25 pendant vertice The s. removal of these 25 pendant vertices yields the path 8 P .The next lemma is easy and its proof is omitted.

Lemma 3.1 A tree is a caterpillar if and only if it does not contain 1,3
K  as a subgraph.

Theorem 3.2 A tree is a permutation graph if an it is a caterpillar. d only if
Proof.A tree that contains 1,3 K  is not a permutation H is equal to a fixed grap h H , we use the symbol  , , , It is easy to check that   a cohesive order of   All graphs of order at most 4 are permutation graphs [1].Therefore,   , , , G P C P P is a permutation graph.Every graph of order n ay be written as If these are the only G sition, then we say that G  Proof.In view of our observation about trees with diameter not exceeding 3, we assume throughout that T has diameter at least 4.
Let T be a tree of order n .Assume that T is a prime permutation graph.By Theorem 3.2 T is a caterpillar.Suppose that 1 x and 2 x are pendant vertices with a common neighbor y .Let G be the tree obtained from T by identifying 1 x and 2 x .Let , , , n y y y   be the vertices of G Without loss of generality, assume that y is the vertex resulting from the id 1 entifi tion of 1 ca x and 2 x .Let 1 H be the grap with two vertices but without an edge, and let i h H be the trivial graph for 2, , 1 dicts the fact that T s prime., , , k G U H H   h and so is itself prime permutation ph or a composition of permutation graphs by a prime th is a permutation grap gra permutation graph.So we see that a permutation gra expressible in terms of prime permutation graphs graphs.

1
Let n    .The graph of inversions of  , denoted by G  , is the graph with vertices where 1, 2, , n  xy is an edge of G  if and only if   , x y or  ,  y x is an inversion of  .

2 , and   5 , 2 .
It is convenient to draw the graph G  with the vertices in a line following their arrangement in  .A drawing of G  is shown in Figure 1.


The mapping  is b e since the scor of the vertices are distinct.It rema ijectiv es ins to show that  In D we ave the arc   , i j v v .Since the tournament T is transitive, then by Theorem 2.2, are adjacent in G  .Conversely, let ab be an edge in G  .Then either  , b a is an inversion.Without loss of   , a b or generality, assum

Using the bottom drawing in Figure 2 ,
we construct a digraph by ecting all edge fr G , ly c om right to left.Then the result is a transitive tournament.It is not difficult to get the score of any vertex in this tournament.We simp count the eastbound arcs and the westbound arcs with a fixed tail.Consider for example, 2 4 v x  .The number of eastbound arcs with tail at 4 Figure 3.

Figure 2 Figure 3 .
Figure 2. A graph with cohesive order because they have cohesive orders as illustrated in Figure 4.

k where 1 k
 all vertices i with 0 i k   are between 0 and k .Moreover, the vertex i is adjacent to 0. Therefore condition (b) is satisfied.Condition (a) is satisfied vacuously.Paths and stars are trees but not all trees are permutation graphs.Consider the tree 1,3 K  formed by subdividing each edge of the star 1,3


 to denote the c The sum of two graphs L and G H omposition. M , denoted by L M  is formed by taking the disjoint union L and of M and then a ding all dges of the f r ab where d

Fi
of U , where V is non-triv l.Since V is a decomposition of U and vertices of U are the induced subgraph contradicts the choice of U .T refore, U must be a prime permutation graph.he4.Concluding RemarksTheore 3.5 is a fair structural escription of a p -m d er matica Academiae Scientiarum Hungarica, Vol. 18, No. 1-2, 1967, pp.25-66.doi:10.1007/BF02020961[4] A. Pnueli, A. Lempel an mutation graph.Each i H in e decomposition Transitiv Orientierbare Graphen," Acta Mathe- H d S. Even, "Transitive Orientation of Graphs and Identification of Permutation Graphs," Canadian Journal of Mathematics, Vol. 23, No. 1, 1971, pp.160-175.doi:10.4153/CJM-1971-016-5ph is by G and two distinct vertices a and b form the edge ab in G if and only if ab is not a edge in G .
is prime.