The Forcing Path Induced Geodetic Number of Some Graphs
J. Arul Asha
1and S. Joseph Robin
2Department of Mathematics,
Scott Christain College, Nagercoil - 629001, INDIA.
email 1: [email protected] 2: [email protected]
(Received on: September 14, 2018)
ABSTRACT
Let G be a connected graph with at least two vertices. A connected geodetic set ⊆ ( ) is said to be a path induced geodetic(pig) set of if < > contais a path , where ( ) = .The minimum cardinality of a path induced geodetic set of is called a path induced geodetic number of and is denoted by ( ). Let be a minimum path induced geodetic set of . A subset is called a forcing subset for if is the unique minimum path induced geodetic containing . A forcing subset for of minimum cardinality is a minimum forcing subset of . The forcing path
induced geodetic number ofdenoted by ( ) is the cardinality of a minimum forcing subset of . The forcing path induced geodetic number of , denoted by is ( ) = min ( ) , where the minimum is taken over all minimum path induced geodetic sets in . The forcing path induced geodetic number of certain classes of graphs are determined. Connected graph of order with forcing path induced geodetic number 0 or 1 are characterised. It is shown that for any positive integer with ≥ 0, there exists a connected graph such that ( ) = =
( ). It is shown that for any positive integer with ≥ 1, there exists a connected graph such that ( ) = and ( ) = .
AMS Subject Classification: 05C12
Keywords: geodesic, geodetic number, connected geodetic number, path induced
geodetic number, forcing path induced geodetic number.
1. INTRODUCTION
By a graph = ( , ), 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
1. The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u v path in G. An u v path of length d(u,v) iscalled an u v geodesic. A vertex x issaid to lie on a u- v geodesic P if x is a vertex of P including the vertices u and v. A vertex is said to be an extreme vertex if the subgraph induced by its neighbours is complete. A of is a set ⊆ ( ) such that every vertex of iscontained in a geodesicjoiningsome pair of vertices in . The ( ) of is the minimum order of its geodetic sets and geodetic set of order ( ) is called − or . The geodetic number of a graph was introduced in
1,2,3,4. A connected geodetic set of a graph G is a geodetic set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected geodetic set of G is the connected geodetic number of G and is denoted by g
c(G). A connected geodetic set of cardinality g
c(G) is called a g
c-set of G or a connected geodetic basis of G. The connected geodetic number of a graph was introduced in
6,7. A connected geodetic set ⊆ ( ) is said to be a path induced geodetic(pig) set of if < > contais a path with ( ) = .The minimum cardinality of a path induced geodetic set of is called a path induced geodetic number of and is denoted by ( ). The path induced geodetic number of a graph was introduced in
5. The following theorems are used in sequel.
Theorem1.1[5].Each extreme vertex of a graph connected belongs to every path induced geodetic set of G. In particular, each end-vertex of belongs to every path induced geodetic set of .
Theorem1.1[5]. Every cut vertex of a connected graph belongs to every path induced geodetic set of .
2. THE FORCING PATH INDUCED GEODETIC NUMBER OF SOME GRAPHS
Definition 2.1. Let G be a connected graph ans be a minimum path induced geodetic set of . A subset is called a forcing subset for if is the unique minimum path induced geodetic set containing . A forcing subset for of minimum cardinality is a minimum forcing subset of . The forcing path induced geodetic number of denoted by ( ) is the cardinality of a minimum forcing subset of . The forcing path induced geodetic number of , denoted by ( ),is ( ) = min ( ) , where the minimum is taken over all minimum path induced geodeticsets in .
Example 2.2. For the graph given in Figure 2.1, = { , , , } and = { , , , } are the only two pig-sets of such that ( ) = ( ) = 1 so that
( ) = 1.
Figure 2.1
Remark 2.3. For the graph given in Figure 2.1, = { , , , }, = { , , , } and = { , , , } are − sets of such that ( ) = ( ) = ( ) = 2 so that ( ) = 2. Thus the forcing path induced geodetic number of a graph and the forcing connected geodetic number of a graph are different.
The next theorem follows immediately from the definition of the path induced geodetic number and the forcing path induced geodetic number of a connected graph . Theorem2.4. For every connected graph, 0 ≤ ( ) ≤ ( ).
The following theorem characterizes graphs for which the bounds in Theorem2.3 are attained and also graphs for which ( ) = 1.The proof of the theorem is sraight forward so we omit the proof.
Theorem2.5. Let be a connected graph. Then
(i) ( ) = 0 if and only if G has a unique minimum path induced geodetic set.
(ii) ( ) = 1 if and only if G has at least two minimum path induced geodetic sets, one of which is a unique minimum path induced geodetic set containing one of its elements, and (iii) ( ) = ( ) if and only if no minimum path induced geodetic set of Gis the unique minimum path induced geodeticset containing any of its proper subsets.
Definition2.6. A vertex v of a graph G is said to be a path induced geodetic vertex if v belongs to every minimum path induced geodetic set of G.
Example 2.7. For the graph G given in Figure 2.1, = { , , , } and = { , , , } are the only two pig-sets of so that ,
and are path induced geodetic vertices of G.
Theorem2.8. Let be a connected graph and be the set of all path induced geodetic vertices of . Then
( ) ≤ ( ) – | |.Proof. Let be any minimum path induced geodetic set of . Then ( ) = | |,
and S is the unique minimum path induced geodetic set containing – . Thus ( ) ≤ | – | = | | – | | = ( ) – | |. ∎
Theorem 2.9. For the complete graph = ( ≥ 2) or a path = , ( ) = 0.
Proof. For = , it follows from Theorem 1.1 that the set of all vertices of G is the unique path induced geodeticset of . Hence it follows from Theorem 2.5(i) that ( ) = 0. If
= , then by Theorem1.1, the set of all end-vertices of Gis the unique minimum path induced geodeticset of G and so ( ) = 0 by Theorem 2.5(i). ∎
Theorem 2.10. If G is a connected graph with ( ) = 2, then ( ) = 0.
Proof. If ( ) = 2, then = . By Theorem 2.5(i), ( ) = 0. ∎ Theorem 2.11. For the cycle
= ( ≥ 4), ( ) = ( ) = + 1 is even + 2 if is odd.
Proof. Let be even and = 2 . Let ∶ , , , . . . , , be the cycle of order 2 . Then is the antipodal vertex of . Let = { , }. Then is a geodetic set of . It is clear that < > is not connected. But = { , , . . . , } is a connected geodetic set of .Since < > contais a path with ( ) = , path induced geodetic set of so that ( ) ≤ + 1. If is any path induced geodetic set of vertices of with | | < | |, then contains at most elements. Hence no two vertices of are pairwise antipodal. Thus is not a geodetic set of . It follows that ( ) = + 1. We prove that = + 1. We have = { , , . . . , } is a − set of . It is easily verified that any subset of with | | < | | is not a forcing subset of and so ( ) ≥ + 1. Since is the unique − set of containing { , , . . . , }, it follows that ( ) = + 1. This is true for all − sets of . Therefore ( ) = + 1.
Let be odd and = 2 + 1. Let : , , . . . , , be the cycle of order 2 + 1. Then and are antipodal vertices of . Let = { , , }. It is clear that is a geodetic set of and < > is not connected. But = { , , … , , , } is a path induced geodetic set of so that ( ) ≤ + 2. is a connected geodetic set of . Since < > contais a path with ( ) = , path induced geodetic set of so that ( ) ≤ + 2. If is any path induced geodetic set of vertices of with | | < | |, then contains at most + 1 elements. Hence contains at most 2 vertices say and which are antipodal to each other. Let be the antipodal vertex of . Then the vertex does not lie on any geodesic joining a pair of vertices of thus is not a geodetic set of . It follows that ( ) = + 2 = + 2. We prove that =
+ 2. We have = { , , … , , , } is a − set of . It is easily verified that
any subset of with | | < | | is not a forcing subset of and so ( ) ≥ + 1. Since
is the unique − set of containing { , , … , , , } , it follows that
( ) = + 2. This is true for all − sets of . Therefore ( ) = + 2. ∎
Theorem2.12. For any positive integer with ≥ 3, there exists a connected graph such that ( ) = = ( ).
Proof. Let = ( ≥ 3). Then by Theorem 2.11, ( ) = = ( ). ∎ Theorem2.13. For any positive integer with ≥ 0, there exists a connected graph such that
( ) = and ( ) = 2 + 1.
Proof. If = 0, let = .Then by Theorem 2.9 and by Theorem 1. 1, ( ) = 0 and ( ) = 2 + 1. Now, assume that ≥ 1. Let ( ) = { , , … , }. Let be the graph obtained from ( ) by adding new vertices , , … , and
,, . . and join each and with and (1 ≤ ≤ ).The graph shown in Figure 2.2. Let = { , , … , } be the set of cut vertices of . Then by Theorem 1.1, every path induced geodetic set of G contains . Since < > is not connected the, is not a path induced geodetic set of . Since is the antipodal vertex of , every minimum path induced geodetic set of G contains { , }. It is easily verified that = ∪ { , } is not a path induced geodetic set of . Let = { , }(1 ≤ ≤ ). It is easily observed that every - set of contains exactly one vertex of each (1 ≤ ≤ )and so ( ) ≥ 2 + 1. Let = ∪ { , , … , }. Then is a - set of G so that ( ) = 2 + 1.
Next, we show that ( ) = . Since every path induced geodetic set of G contains it follows from Theorem 2.8 that ( ) ≤ ( ) – | | = . Now, since ( ) = 2 + 1 and every - set of contains , it is easily seen that every -set is of the form ∪ { , , … , }, where (1 ≤ ≤ ). Let be any proper subset of with | | < . Then there is a vertex (1 ≤ ≤ ) suchthat . Let be a vertex of distinct from . Then = ( – { }) ∪ { } is a -set properly containing . Thus is not the unique -set containing and so is not a forcing subset of . This is true for all
-sets of and so ( ) = . ∎
Figure 2.2
