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These are the Two-free Trees
Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity J. NeSetfil and M. Fiedler (Editors) 0 1992 Elsevier Science Publishers B.V. All rights reserved.
These are t h e Two-free Trees J. SHEEHAN A N D C. R . J . CLAPHAM
Using the definition of k-free, a known result can be re-stated as follows: If G is not edge-reconstructible then G is k-free, for all even k. It is known that trees are edge-reconstructible; but an alternativeproof of this can be obtained by combining the result above with the result outlined here that, apart from paths, all trees (except a finite number, which are determined) are not 2-free. The approach may be of use in obtaining further results on edge-reconstruction.
1. Introduction
T h e graphs in this paper are connected simple graphs with n vertices. Such a graph G will be considered as a spanning subgraph of I<,. T h e following is a definition of ‘k-free’, though the way that it is used will be seen more clearly below. Indeed, we recommend the reader to take, as the definition to work with, the property given later in Lemma 2.
Definition. Suppose that G is a graph and that 1 6 k 6 IE(G)(.Then G is k-free if, for every subset A o f E ( G )with 1Al = IE(G)I-k, there exists an automorphism p of I<, such that E ( G ) n E(p(G)) = A . A graph is even-free if it is k-free, for all even k.
Also the following result is known (see Nash-Williams [ 2 ] ) : Lemma 1. If G is not edge-reconstructible then, for every subset A of E ( G ) such that IAl G IE(G)I (mod a ) , there exists an autornorphism p ofK, such that E(G)n E(P(G)) = A . Now this lemma says that if a graph is not edge-reconstructible then it is even-free, and so a graph that is not k-free, for some even value of k , is edgereconstructible. An investigation into which graphs are, or are not, k-free may therefore bring closer the settling of the Edge-Reconstruction Conjecture. For example, it is known that trees are edge-reconstructible. An alternative method of establishing this is to show t h a t , apart from paths, all trees (except a finite number, which are determined) are not 2-free. For then, paths and the finite number of other trees t h a t are 2-free can clearly be shown to be edge-reconstructible; and the others, being not 2-free, are also edge-reconstructible. Here we shall outline 309
310
J . Sheehan and C. R. J. C l a p h a m
a proof that the only 2-free trees are those shown in Figure 2. It has to be admitted that a complete account involves a lot of detail.
6
FIGURE 1
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FIGURE 2
The definition above was presented in [4]and an investigation into which graphs are k-free, for all k, was begun in [l]. It was shown in [3] that, apart from paths, the only 1-free trees are the two shown in Figure 1. The following method is used in practice to determine whether or not a graph is k-free. Let { e l , e 2 , . . . , e k } be a set of edges of a graph G. A replacing set is any set of edges that can be added to G - { e l , e 2 , . . . , e k } to form a graph isomorphic t o G. The replacing set { f l , f 2 , . . . , fk} is called a disjoznt replacing set if { e 1 , e 2 , . . . , e k } n { f l , f 2 , . . . , f k } = 8. If there is a disjoint replacing set for { e 1 , e 2 , . . . , e k } , we shall say that the set { e l , e z , . . . , e k } is replaceable. The following is immediate. Lemma 2. The graph G is k-free if and only if every set of k edges is replaceable.
So we shall outline a proof of the following: Theorem. Apart from paths, the only 2-free trees are those shown in Figure 2 The method will be to show that for all other trees it is possible to find a pair of edges { e l , e 2 } that is not replaceable.
T h e s e are t h e Two-free Trees
31 1
2. Sketch of the Proof I t is known t h a t a tree T has either one or two centres. (We may also refer t o a centre as a central vertex.) Let 2 denote the induced subgraph of T whose vertex-set V(2)consists of the centre(s) of T . We define S , the s p i n e of T , t o be the induced subgraph whose vertex-set V ( S )consists of those vertices that belong t o every longest path of T ; thus, S is the intersection of all longest paths. Then S is a path and V ( S )contains the centre(s). If v E V ( S ) ,v is a spinal vertex. Now let us define the spinal distance spd(v) of a vertex v by spd(v) = niin{d(v, s)
I s E V(S)}.
This is the distance of v from the spine. Then we shall call T a spindly tree if spd(v) 6 1 for all v, except for a number of ‘permissible’ vertices with spinal distance 2. [The precise meaning of permissible is this: Suppose t h a t spd(v) = 2. Let s be the spinal vertex distance 2 from v and let CY be the centre nearest t o s. Then v is permissible if either s = CY or CY and all the vertices between s and CY have degree 2.1 T h e proof we are outlining can be broken into two parts. T h e first step about which more will be said below is t o show t h a t if 7’ is not a spindly tree then a pair of edges t h a t is not replaceable can be found. T h e second step consists of looking a t spindly trees and showing t h a t , again, a non-replaceable pair of edges can be found unless the tree is one of those in Figure 2. T h e details of this step will not be given here. T h e first step, which proves that all 2-free trees are spindly, has t o explain how, if the tree is not spindly, a non-replaceable pair of edges can be found. T h e first edge e l is a pendant edge ( t h a t is, incident with a vertex of degree 1) chosen in a very particular way; it is the end edge of a ‘critical’ path, whose definition is the subject of the next section. T h e second edge e2 is a non-pendant edge, which can be chosen much more freely; all that is required in the choice of e 2 , roughly, is that when e l and en are removed, the new tree has the same centre(s) as the original. 3. Critical Path
T h e purpose then of this section is t o arrive a t the definition of critical path. For any non-central vertex v, let us define the predecessor v-: it is the unique vertex adjacent t o v and belonging t o the component of T - u that contains 2. Thus, if a is a centre, d ( a , v-) = d ( a , v) - 1. The definition of a successor will also be needed: For any non-central vertex u , let N + ( v ) = N ( v ) \ {v-}. If v+ E N+(’u), v+ is a successor of v. Thus, if a is a centre and v+ is a successor of v , d ( a ,v+) = d ( a , v) 1. For a central vertex a , let N + ( a ) = N ( a ) \ V ( 2 ) .In fact, the successors of 2, are precisely those vertices that have v as predecessor. We’ll also need this notation: For any non-central vertex v , let T, be the component of T - v - that contains ’u. For central vertices, the appropriate definition is: If V(2)= { a , / ? }let , T, be the component of T - /? that contains a , and Tp the component of T - a that contains /?. If V(2)= { a } ,let T, = T.
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J . Sheehan and C. R. J . Clapham
Define now the weight ~ ( v of ) a vertex v by W ( W ) = min{lV(Tu)l: u E N+(w)\ V ( S ) } ,if N + ( v ) V ( S ) (that is, if v has a non-spinal successor), and w ( v ) = IV(Tv+)l,if N + ( v ) = {v+} C V ( S ) (which occurs only if v is spinal and has just one successor v+). Now let us call a path in T centrifugal (defined in the dictionary as ‘tending away from the centre’) if it is defined by vertices vo, v1, . . . , v,, where s 2 1, vo E V(Z), each vi is a successor of vi-1, and deg(v,) = 1. For such a centrifugal path P , let A ( P ) be defined by
Thus A(P) is the length of the part that P has in common with the spine. Now let
AT = min{A(P): P is a centrifugal path of T } . It follows t h a t AT is the minimum distance from a centre t o a vertex of degree 1 or 2 3. (If a centre has degree 2 3, AT = 0.) For any path P in T , defined by vertices v o l . . . Or, where T 2 1, vo E V ( 2 ) and each vi is a successor of w i - 1 (with vr not necessarily of degree l ) , define the weight sequence w( P ) by
We shall want to compare weight sequences and this will be done by defining the following ordering. Let w = (wo,w1, w2,.. .) and x = ( X O ,~ 1 ~ x 2. ,.) .be sequences of non-negative integers (with finitely many non-zero). Write w < x if, for some i, and w,< x i . w 1 = X I , . . . , w;-1 = So we arrive at the definition of a critical path as a centrifugal path with certain minimality conditions:
Definition. A centrifugal path P , defined by vertices v o , . . . , v,, is crzlzcal if (i) A ( P ) = A T , (ii) For i = 1, . . ., s , IV(T,,)I = w(vi-l),and vi $ V(S) i f N + ( u i - l ) 9 V ( S ) (that is, v, is a successor of vi-1 that is non-spinal (unless v i - 1 is spinal with only one successor).),with IV(Tv,)las small as possible for all such), (iii) subject to (i) and (ii), the weight sequence w ( P ) , is minimal with respect to <, (iv) subject to (i).),(ii) and (iii), the number of vertices of degree 1 adjacerit to v,-1 is minimal. It follows t h a t , if P is a critical path and Q is any centrifugal path, then w(P) 6 w(Q); and if w(P) = w(Q), then P and Q have the same length and the number of vertices of degree 1 adjacent t o the penultimate vertex of P is less than or equal t o the number of vertices adjacent t o the penultimate vertex of Q .
These are the Two-free Trees
313
4. Conclusion
So now we take an end edge of a critical path P as el and some non-pendant edge, about which there is some choice, as e z . This is possible if T is not spindly. Suppose that the pair of edges { e l , e z } is replaced by a disjoint pair { f l , fz}. Let T’be defined by V(T’) = V ( T )and E(T’) = E ( T ) - { e I , e z } {fi,fz}. We find that in T‘ either there are fewer critical paths than there are in T , or there is a path in T’ t h a t , if T’ were isomorphic t o T, would contradict the criticality of P . Consequently { e l , e z } is a non-replaceable pair, as required.
+
5. Conjecture
I t can be seen immediately t h a t the 1-free trees are 2-free. Further, it can be shown that the 2-free trees are 3-free. Indeed, for the small trees t h a t we have investigated, it appears that if a tree with R vertices is L-free then it is (k 1)-free (for k < R - 2). It would be interesting t o find out whether this result is true for all trees.
+
References
[l] C. R. J . Clapham and J . Sheehan, Super-free Graphs, Ars Combinatoria, t o appear. [a] C. S t J . A. Nash-Williams, The Reconstruction Problem, Selected Topics in Graph Theory, Academic Press, 1978. [3] J . Sheehan, Fizzng Subgraphs, J . Combin. Theory (B) 12 (1972), 226-243. [4]J . Sheehan and C. R. J . Clapham, Edge-reconstructzon and k-free graphs, Utilitas (Proceedings of the British Combinatorial Conference), 1989, t o appear.
J . Sheehan and C. R. J . Clapham Department of Mathematical Sciences, University of Aberdeen, Aberdeen, U K
These are t h e Two-free Trees J. SHEEHAN A N D C. R . J . CLAPHAM
Using the definition of k-free, a known result can be re-stated as follows: If G is not edge-reconstructible then G is k-free, for all even k. It is known that trees are edge-reconstructible; but an alternativeproof of this can be obtained by combining the result above with the result outlined here that, apart from paths, all trees (except a finite number, which are determined) are not 2-free. The approach may be of use in obtaining further results on edge-reconstruction.
1. Introduction
T h e graphs in this paper are connected simple graphs with n vertices. Such a graph G will be considered as a spanning subgraph of I<,. T h e following is a definition of ‘k-free’, though the way that it is used will be seen more clearly below. Indeed, we recommend the reader to take, as the definition to work with, the property given later in Lemma 2.
Definition. Suppose that G is a graph and that 1 6 k 6 IE(G)(.Then G is k-free if, for every subset A o f E ( G )with 1Al = IE(G)I-k, there exists an automorphism p of I<, such that E ( G ) n E(p(G)) = A . A graph is even-free if it is k-free, for all even k.
Also the following result is known (see Nash-Williams [ 2 ] ) : Lemma 1. If G is not edge-reconstructible then, for every subset A of E ( G ) such that IAl G IE(G)I (mod a ) , there exists an autornorphism p ofK, such that E(G)n E(P(G)) = A . Now this lemma says that if a graph is not edge-reconstructible then it is even-free, and so a graph that is not k-free, for some even value of k , is edgereconstructible. An investigation into which graphs are, or are not, k-free may therefore bring closer the settling of the Edge-Reconstruction Conjecture. For example, it is known that trees are edge-reconstructible. An alternative method of establishing this is to show t h a t , apart from paths, all trees (except a finite number, which are determined) are not 2-free. For then, paths and the finite number of other trees t h a t are 2-free can clearly be shown to be edge-reconstructible; and the others, being not 2-free, are also edge-reconstructible. Here we shall outline 309
310
J . Sheehan and C. R. J. C l a p h a m
a proof that the only 2-free trees are those shown in Figure 2. It has to be admitted that a complete account involves a lot of detail.
6
FIGURE 1
__%_%__t%%__o
FIGURE 2
The definition above was presented in [4]and an investigation into which graphs are k-free, for all k, was begun in [l]. It was shown in [3] that, apart from paths, the only 1-free trees are the two shown in Figure 1. The following method is used in practice to determine whether or not a graph is k-free. Let { e l , e 2 , . . . , e k } be a set of edges of a graph G. A replacing set is any set of edges that can be added to G - { e l , e 2 , . . . , e k } to form a graph isomorphic t o G. The replacing set { f l , f 2 , . . . , fk} is called a disjoznt replacing set if { e 1 , e 2 , . . . , e k } n { f l , f 2 , . . . , f k } = 8. If there is a disjoint replacing set for { e 1 , e 2 , . . . , e k } , we shall say that the set { e l , e z , . . . , e k } is replaceable. The following is immediate. Lemma 2. The graph G is k-free if and only if every set of k edges is replaceable.
So we shall outline a proof of the following: Theorem. Apart from paths, the only 2-free trees are those shown in Figure 2 The method will be to show that for all other trees it is possible to find a pair of edges { e l , e 2 } that is not replaceable.
T h e s e are t h e Two-free Trees
31 1
2. Sketch of the Proof I t is known t h a t a tree T has either one or two centres. (We may also refer t o a centre as a central vertex.) Let 2 denote the induced subgraph of T whose vertex-set V(2)consists of the centre(s) of T . We define S , the s p i n e of T , t o be the induced subgraph whose vertex-set V ( S )consists of those vertices that belong t o every longest path of T ; thus, S is the intersection of all longest paths. Then S is a path and V ( S )contains the centre(s). If v E V ( S ) ,v is a spinal vertex. Now let us define the spinal distance spd(v) of a vertex v by spd(v) = niin{d(v, s)
I s E V(S)}.
This is the distance of v from the spine. Then we shall call T a spindly tree if spd(v) 6 1 for all v, except for a number of ‘permissible’ vertices with spinal distance 2. [The precise meaning of permissible is this: Suppose t h a t spd(v) = 2. Let s be the spinal vertex distance 2 from v and let CY be the centre nearest t o s. Then v is permissible if either s = CY or CY and all the vertices between s and CY have degree 2.1 T h e proof we are outlining can be broken into two parts. T h e first step about which more will be said below is t o show t h a t if 7’ is not a spindly tree then a pair of edges t h a t is not replaceable can be found. T h e second step consists of looking a t spindly trees and showing t h a t , again, a non-replaceable pair of edges can be found unless the tree is one of those in Figure 2. T h e details of this step will not be given here. T h e first step, which proves that all 2-free trees are spindly, has t o explain how, if the tree is not spindly, a non-replaceable pair of edges can be found. T h e first edge e l is a pendant edge ( t h a t is, incident with a vertex of degree 1) chosen in a very particular way; it is the end edge of a ‘critical’ path, whose definition is the subject of the next section. T h e second edge e2 is a non-pendant edge, which can be chosen much more freely; all that is required in the choice of e 2 , roughly, is that when e l and en are removed, the new tree has the same centre(s) as the original. 3. Critical Path
T h e purpose then of this section is t o arrive a t the definition of critical path. For any non-central vertex v, let us define the predecessor v-: it is the unique vertex adjacent t o v and belonging t o the component of T - u that contains 2. Thus, if a is a centre, d ( a , v-) = d ( a , v) - 1. The definition of a successor will also be needed: For any non-central vertex u , let N + ( v ) = N ( v ) \ {v-}. If v+ E N+(’u), v+ is a successor of v. Thus, if a is a centre and v+ is a successor of v , d ( a ,v+) = d ( a , v) 1. For a central vertex a , let N + ( a ) = N ( a ) \ V ( 2 ) .In fact, the successors of 2, are precisely those vertices that have v as predecessor. We’ll also need this notation: For any non-central vertex v , let T, be the component of T - v - that contains ’u. For central vertices, the appropriate definition is: If V(2)= { a , / ? }let , T, be the component of T - /? that contains a , and Tp the component of T - a that contains /?. If V(2)= { a } ,let T, = T.
+
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J . Sheehan and C. R. J . Clapham
Define now the weight ~ ( v of ) a vertex v by W ( W ) = min{lV(Tu)l: u E N+(w)\ V ( S ) } ,if N + ( v ) V ( S ) (that is, if v has a non-spinal successor), and w ( v ) = IV(Tv+)l,if N + ( v ) = {v+} C V ( S ) (which occurs only if v is spinal and has just one successor v+). Now let us call a path in T centrifugal (defined in the dictionary as ‘tending away from the centre’) if it is defined by vertices vo, v1, . . . , v,, where s 2 1, vo E V(Z), each vi is a successor of vi-1, and deg(v,) = 1. For such a centrifugal path P , let A ( P ) be defined by
Thus A(P) is the length of the part that P has in common with the spine. Now let
AT = min{A(P): P is a centrifugal path of T } . It follows t h a t AT is the minimum distance from a centre t o a vertex of degree 1 or 2 3. (If a centre has degree 2 3, AT = 0.) For any path P in T , defined by vertices v o l . . . Or, where T 2 1, vo E V ( 2 ) and each vi is a successor of w i - 1 (with vr not necessarily of degree l ) , define the weight sequence w( P ) by
We shall want to compare weight sequences and this will be done by defining the following ordering. Let w = (wo,w1, w2,.. .) and x = ( X O ,~ 1 ~ x 2. ,.) .be sequences of non-negative integers (with finitely many non-zero). Write w < x if, for some i, and w,< x i . w 1 = X I , . . . , w;-1 = So we arrive at the definition of a critical path as a centrifugal path with certain minimality conditions:
Definition. A centrifugal path P , defined by vertices v o , . . . , v,, is crzlzcal if (i) A ( P ) = A T , (ii) For i = 1, . . ., s , IV(T,,)I = w(vi-l),and vi $ V(S) i f N + ( u i - l ) 9 V ( S ) (that is, v, is a successor of vi-1 that is non-spinal (unless v i - 1 is spinal with only one successor).),with IV(Tv,)las small as possible for all such), (iii) subject to (i) and (ii), the weight sequence w ( P ) , is minimal with respect to <, (iv) subject to (i).),(ii) and (iii), the number of vertices of degree 1 adjacerit to v,-1 is minimal. It follows t h a t , if P is a critical path and Q is any centrifugal path, then w(P) 6 w(Q); and if w(P) = w(Q), then P and Q have the same length and the number of vertices of degree 1 adjacent t o the penultimate vertex of P is less than or equal t o the number of vertices adjacent t o the penultimate vertex of Q .
These are the Two-free Trees
313
4. Conclusion
So now we take an end edge of a critical path P as el and some non-pendant edge, about which there is some choice, as e z . This is possible if T is not spindly. Suppose that the pair of edges { e l , e z } is replaced by a disjoint pair { f l , fz}. Let T’be defined by V(T’) = V ( T )and E(T’) = E ( T ) - { e I , e z } {fi,fz}. We find that in T‘ either there are fewer critical paths than there are in T , or there is a path in T’ t h a t , if T’ were isomorphic t o T, would contradict the criticality of P . Consequently { e l , e z } is a non-replaceable pair, as required.
+
5. Conjecture
I t can be seen immediately t h a t the 1-free trees are 2-free. Further, it can be shown that the 2-free trees are 3-free. Indeed, for the small trees t h a t we have investigated, it appears that if a tree with R vertices is L-free then it is (k 1)-free (for k < R - 2). It would be interesting t o find out whether this result is true for all trees.
+
References
[l] C. R. J . Clapham and J . Sheehan, Super-free Graphs, Ars Combinatoria, t o appear. [a] C. S t J . A. Nash-Williams, The Reconstruction Problem, Selected Topics in Graph Theory, Academic Press, 1978. [3] J . Sheehan, Fizzng Subgraphs, J . Combin. Theory (B) 12 (1972), 226-243. [4]J . Sheehan and C. R. J . Clapham, Edge-reconstructzon and k-free graphs, Utilitas (Proceedings of the British Combinatorial Conference), 1989, t o appear.
J . Sheehan and C. R. J . Clapham Department of Mathematical Sciences, University of Aberdeen, Aberdeen, U K