Trees with distinguishing number two
The distinguishing number of a graph is the least integer such that has a vertex labeling with labels that is preserved only by a trivial automorphism. In this paper we characterize all trees with radius at most three and distinguishing number two. Also we present a necessary condition for trees with distinguishing number two and radius more than three.
Department of Mathematics, Yazd University, 89195-741, Yazd, Iran
Keywords: distinguishing number; tree; radius.
AMS Subj. Class.: 05C25
1 Introduction and definitions
Let be a simple connected graph. The automorphism group and the complement of , is denoted by and , respectively. The distance between two vertices of , denoted by , is the length of a shortest path if any; otherwise, . The eccentricity of a vertex in is the distance between and a vertex farthest from . A vertex is an eccentric vertex of a vertex in if . The minimum eccentricity among the vertices of is the radius of , , while the maximum eccentricity is its diameter . A vertex of is called a central vertex if , and center of is the collection of all such vertices in . Graph with is called self-centered graph. Equivalently, a graph is selfcentered if all its vertices lie in the center. Thus, the eccentric set of a self-centered graph contains only one element, that is, all the vertices have the same eccentricity. It follows that any disconnected graph is self-centered graph with radius . The subgraph induced by center of , , in is denoted by . A graph is called -self-centered if . The terminology -equi-eccentric graph is also used by some authors. For studies on these graphs see [2, 4, 5] and .
Distinguishing labeling was first defined by Albertson and Collins  for graphs. A labeling of a graph , , is said to be -distinguishing if no nontrivial automorphism of preserves all the vertex labels. In other words, is -distinguishing if for any , , there is a vertex such that . The distinguishing number of a graph is defined as
It is immediate that for the complete graph of order , and that for , where is the path graph of order . Cheng in  has presented -time algorithms that compute the distinguishing numbers of trees and forests. The distinguishing number of graphs have been extensively studied in the literature. The following general problem appeared in :
Characterize graphs with distinguishing number two.
In this paper, we obtain all trees with radius at most two and distinguishing number two. Also we state a note on the trees of radius greater than two and finally pose a problem.
2 Main results
To obtain all trees with radius at most two and distinguishing number two, we need some preliminaries:
Let be a 2-self-centered graph. A vertex in is called critical for and if and is the only common neighbour of and .
Let be a graph. A cycle in is said to be locally geodesic at a vertex if for each vertex on , the distance between and in coincides with that in .
 Let be a bipartite graph. Then the following three are equivalent:
is isomorphic to the complete bipartite graph ;
is a block and for each vertex of , there is no cycle locally geodesic at and of length more than .
Now, we can consider edge-minimal 2-self-centered graphs with some triangles. We need the following procedure to proceed.
Procedure. Let be a graph, form a triangle in and suppose that is a critical vertex for and and/or is a critical vertex for and . Remove the edge and add edges and . The following theorem characterizes edge-minimal -self-centered graphs with triangles, which completes the characterization of all 2-self-centered graphs.
 Let be a graph. Then is an edge-minimal -self-centered graph with some triangle if and only if the following two conditions are true:
For each edge of every triangle in , at least one end-vertex is a critical vertex (for the other end-vertex of that edge and some other vertices of ), and
Iteration of above Procedure on (at most to the number of triangles of ) transforms to a triangle-free -self-centered graph.
The sequential join , of graphs , is formed from , by adding the additional edges joining each vertex of with each vertex of , . The double star is the tree , and the tritip is the graph formed by adding , and pendant edges at the vertices of , see .
The Petersen graph,
The graph formed from by adding an additional vertex and joining to each vertex of degree 1 in .
The graph formed from by adding a new vertex and joining to each vertex of degree 1 in , , .
 Let be a tree. Then, if and only if .
 Let be a tree of order . Then, is either or isomorphic to .
Let be a tree of radius at most two. The distinguishing number of is two if and only if is one of the trees in Figure 2.
Proof. We know that given an unrooted tree we can construct a rooted tree such that . It is well known that a tree either has one center (i.e., it is unicentral) or has two adjacent centers (i.e., it is bicentral). Thus, if has a unique center, simply let be a copy of ; otherwise, let be the tree formed by appending a new vertex to the two centers of and deleting the edge between the two old centers of . In both cases, has a unique center which we designate as its root . So for obtaining all trees with radius at most two and the distinguishing number two it is sufficient to consider all rooted trees of radius at most two with distinguishing number two. Only with respect to the degree of the root, we can get these trees as shown in Figure 2. Note that the roots have shown in red colour.
Here, we state a note about trees with the distinguishing number two and radius more than two. First we state and prove the following theorem:
There is no tree with diameter tree, radius and .
Proof. Since , so the subgraph induced by center of , , in , is by Corollary 2.7. So the only possible number for for which is , and hence . But , and therefore there is no tree with distinguishing number two in this case.
If is a tree with radius and , then is a connected -self-centered graph that is not bipartite.
Proof. If is a tree of order with radius and , then the radius of the complement of is . By Theorem 2.9, , and then by Theorem 2.6. It is known that the complement of a tree is connected or it is a union of an isolated vertex and a complete graph. If is a union of an isolated vertex and a complete graph, then is a star graph, and since so . But , thus there is no tree with distinguishing number two in this case. Therefore is a connected -self-centered graph that is not bipartite by Corollary 2.3 and Theorem 2.6.
By Theorem 2.10, for finding all trees with and , we should only consider the -self-centered trees that are not bipartite. But Buckley in  obtained all -self-centered graphs having as few edges as possible among such graphs. Since is a tree, so is not an edge-minimal -self-centered graph by Theorem 2.4. Thus we should add a specific number of edges to each edge-minimal -self-centered graph so that we get . We end this paper with the following problem:
Characterize edges which add to each edge-minimal -self-centered graph such that .
- Corresponding author
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