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32URAL MATHEMATICAL JOURNAL, Vol. 7, No. 2, 2021, pp. 121-135
DOI: 10.15826/umj.2021.2.009
ON HOP DOMINATION NUMBER OF SOME GENERALIZED GRAPH STRUCTURES
S. Shanmugavelan^, C. Natarajan^
Srinivasa Ramanujan Centre, SASTRA Deemed to be University, Kumbakonam-612001, India
tshanmugavelan@src.sastra.edu, ttnatarajan_c@maths. sast.ra.edu
Abstract: A subset H C V(G) of a graph G is a hop dominating set (HDS) if for every v € (V \ H) there is at least one vertex u € H such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called the hop domination number of G and is denoted by yh(G). In this paper, we compute the hop domination number for triangular and quadrilateral snakes. Also, we analyse the hop domination number of graph families such as generalized thorn path, generalized ciliates graphs, glued path graphs and generalized theta graphs.
Keywords: Hop domination number, Snake graphs, Theta graphs, Generalized thorn path.
1. Introduction
Domination in graphs is fascinating topic in the field of graph theory. It is one of the most effective mathematical models for a variety of real world problems. A simple undirected finite graph G holds a vertex set V(G) with vertices and an edge set E(G) whose members are unordered pair of vertices called lines or edges of G. The degree of a vertex v, denoted by d(v), is the number of edges that are incident with v and the distance d(u, v) between any two distinct vertices u and v is the length of the shortest path connecting u and v in G. We use the symbol [n] = {1,2,..., n}. For any other graph theory terminology not defined here, we follow [3].
In a graph G, a subset D C V(G) is said to be a dominating set if every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a minimal dominating set of G is the domination number y(G). In the last three decades, several domination parameters have been established and they have been intensively investigated with applications in communication networks, facility location problems, game theory, mathematical chemistry, and so on. For a detailed study on domination concepts, one may refer [8-10].
Ayyasamy et al. [1] defined a new distance-based domination parameter called the hop domination number of a graph G. A subset H C V(G) of a graph G is a hop dominating set (HDS) if for every vertex v not in H, there exists at least one vertex u € H such that d(u, v) = 2. The minimum cardinality of a hop dominating set of G is called the hop domination number of G and is denoted by Yh(G). The hop degree of a vertex v in a graph denoted by dh(v) is the number of vertices at distance= 2 from v. The hop graph H(G) of a graph G is the graph having same vertex set and two vertices u, v are adjacent in H(G) iff dG(u,v) = 2. Also, Ayyasamy et al. [2] obtained some bounds on hop domination number for trees and characterized trees attaining those bounds. Natarajan et al. [13] found characterization results for hop domination number equals other domination parameters like total domination number, connected domination number for several families of graphs. Many scholars have explored this parameter in the years thereafter, leading to novel versions such as connected hop domination, total perfect hop domination, Roman hop domination,
Global hop domination, etc., [11, 12, 15-18, 20]. In 2018, Natarajan et al. [14] discussed hop domination number for some special families of graph like central graph, middle graph and total graph. Recently, Packiavathi et al. [6] obtained the hop domination number of a caterpillar graph Pn(li, l2, • • •, ln) (a caterpillar is a graph obtained from the path by attaching leaves li to ith vertex of the path Pn) and the domination number for some special families of snake graphs which occur as hop graph of Pn(1,1,..., 1) and Pn(2,2,..., 2). We refine their result on caterpillar graph and present an elegant result.
2. Main results
In this section, we study the hop domination number of snake graph families like triangular, alternate triangular, quadrilateral and alternate quadrilateral snakes. In addition, the hop domination number of some generalized structures like generalized theta graphs, generalized thorn paths and generalized ciliates graphs GC(p, q, t) for p = 3 and p = 4 are determined.
Definition 1 [7]. Let li,l2,...ln be n positive integers. Then the thorn graph Gt = Gt(li,l2 ... ln) is obtained from a graph G by attaching li pendant vertices (thorns) to each vertex vi of G, i € [n].
In 2020, Getchial Pon Packiavathi et al. [6] obtained the following result on caterpillar graphs.
Theorem 1 [6]. Yh(Pn (1,1,... 1)) = Yh(Pn (2, 2,... 2)) =
2r,
if n = 2r;
2r + 3, if n = 2r + 1.
First, we observe that the result given in Theorem 1 is wrong. For example, Yh(P4(1,1,1,1)) = 2 whereas from their computations it is 4. So, we refine the result by taking the more generalized version of caterpillar called thorn path Pjn.
Theorem 2. For n > 1,
Yh(Pt) =
n — 1
+ 1, if n = 0,1, 3 (mod 4); + 1, if n = 2 (mod 4).
Proof. Let vi, ■ ■ ■ , vn be the vertices of the central path Pn in P^ (see Fig. 1). Case 1: n = 2 (mod 4). In this case, any Yh-set is of the form
{vi | i = 1 (mod 4), 1 < i < (n — 2)} U {vj | j = 2 (mod 4), 2 < j < (n — 1)} U {vn-i}.
Thus,
and it is easily seen that Therefore,
Yh(Pn) < Yh(Pn) > Yh(Pn) =
+1
+ 1.
+ 1.
Case 2: n = 0,1, 3 (mod 4). In this case, any Yh-set is of the form
{vi | i = 2 (mod 4), 2 < i < (n — 2)} U {vj | j = 3 (mod 4), 3 < i < (n — 1)} U {vn-i}.
Figure 1. Thorny path P^ •
Thus,
and it is easily seen that
Therefore,
Yh (Pi) <
Yh(Pn) > Yh(Pt ) =
□
In [4], Derya Dogan et al. obtained some results for weak and strong domination in thorn graphs. Inspired by their results, we study our parameter namely, hop domination number for thorn rod given in [4], as well as for other generalized graph structures.
{2r, if n = 6r;
2r + 1, if n = 6r + 1;
2r + 2, if n = 6r + s, 2 < s < 5.
Rewriting Lemma 1 in terms of congruence, we have
Yh(Pn ) = <
n
-3.
n
—
L3 J
n
-3.
, if n = 0 (mod 6); + 1, if n = 1, 3, 4, 5 (mod 6); + 2, if n = 2 (mod 6).
Definition 2 [4]. A thorn rod is a graph Pnwhich is obtained by taking a path on n > 2 vertices and attaching (t — 1) leaves, known as thorns, at each of the end of Pn.
V1 V2 V3 V4
Vm
*t-1
Figure 2. Thorn rod Pn t.
Note that P1,t is a star graph K1)t-1.
Theorem 3. Yh(Pn,t) = <
n - 10
L 3 J
n - 10
L 3 J
n - 10
L 3 J
+ 6, if n = 0 (mod 6); + 5, if n = 1, 2, 3, 5 (mod 6); + 4, if n = 4 (mod 6).
Proof. Let us label the vertices of central path Pn as vi, v2 ... vn. Let the leaves or thorns at the vertex v1 be x1, x2,... xt-1 and the thorns at the vertex vn be y1, y2,... yt-1.
From Fig. 2, it is clear that to hop dominate 2(t — 1) leaves and their support vertices, any Yh-set must include the vertices v2, v3, vn-1, vn-2.
Now, the subgraph induced by Pn — {v1,v2, v3, v4, v5, vn-4, vn-3, vn-2, vn-1, vn}, is clearly a path on (n — 10) vertices. By Lemma 1,
Yh(Pn-io) = <
/ n - 10
L 3 J
n - 10
L 3 J
n - 10
L 3 J
, if n = 4 (mod 6); + 1, if n = 0,1, 3, 5 (mod 6); + 2, if n = 2 (mod 6).
and hence Yh(G) = 4 + Yh(Pn-1o). Thus, the result follows.
□
Definition 3. A glued path GP(n,t) is a graph obtained by gluing t copies of a path Pn(n > 2) at a common vertex v such that v is the initial vertex in each copy of Pn.
Theorem 4. Yh(GP(n,t)) = <
L6J n
6
n
L6-
t, if n = 0, 5 (mod 6);
t + 1, if n = 1 (mod 6);
1t - 1 +1, if n = 4 (mod 6);
t + 2, if n = 2, 3 (mod 6).
Proof. Let us arrange the vertices of GP(n, t) row-wise subject to the following conditions:
(i) Place the common vertex in the 1st row R1.
(ii) First vertex of each copy of the path Pn be placed in the 2nd row R2.
(iii) In general, nth vertex of each copy in the (n + 1)th row Rn+i-
From Fig. 3, it is clear that, each row Ri has t vertices except the first row. That is, V(Rj) = [vi, vj,..., v(t)}, 2 < i < (n + 1). To hop dominate all those leaves and support vertices, all the vertices in Rn-1 and Rn-2 must be selected from a Yh-set of GP(n,t). This choice will also hop dominate all of the vertices of Rn-3 and Rn_4.
rows
Case 1: n = 0 (mod 6). In this case, any Yh-set contains 2[n/6]t vertices from the following
R = [R(n-1), R(n-2), R(n-7), R(n-8), . . . , R4, R3}. Thus,
Yh(GP(n,t)) < 2
t.
It is easy to observe that any hop dominating set of GP(n, t) contains at least 2[n/6]t vertices. Therefore,
n
lh(GP(n,t))=2 - t.
6
Case 2: n = 5 (mod 6). In this case, any Yh-set includes 2[n/6]t vertices from the rows (R\ R3) U[vi}. Thus,
r n 1
t.
Also,
Therefore,
Yh(GP(n,t)) < 2 Yh(GP(n,t)) > 2 Yh(GP (n,t)) =2
t.
Case 3: n = 1 (mod 6). In this case, any Yh-set includes vertices from the following rows
R = [R(n-1) ,R(n-2), R(ra-7), R(n-8), . . . , R5, R4 }u[v1}. Thus,
Yh(GP(n,t)) < 2
n
L6J
t + 1.
t
One can observe that Therefore,
Yh(GP(n,t)) > 2
n
L6-
n
L6-
t + 1.
>yh(GP(n,t))=2[^\t + l. Similarly, the proof follows for other cases. □
The generalized thorn path can be defined as follows,
Definition 4. The graph obtained by taking a path Pn and attaching t copies of Pr to every vertex of Pn is said to be a generalized thorn path and denoted by G(n,r, t),n > 1.
7h(Pn) + nt - , if r ee 0 (mod 6);
Theorem 5. Yh[G(n, r, t)]=<
n + nt r
r
L3_
nt
L3-
if r = 1, 2, 3 (mod 6); + 1) , if r = 4, 5 (mod 6).
Proof. Let S = {vj : 1 < i < r, 1 < j < t} denote the vertices of the ith copy of Pr as shown in Fig. 4.
Figure 4. Generalized thorn path G(n, r, t).
Case 1: r = 0 (mod 6). In this case, the set
= {v(n-2)j, v(n-1)j, v(n-8)j, v(n-9)j • • • v4j, v3j, 1 < j < t}
hop dominates all vertices in each copy of Pr. In order to hop dominate the vertices of the path Pn, any Yh-set of G(n,r, t) should contains Yh(Pn) vertices. As a result,
It is easily seen that
Yh[G(n, r, t)] < Yh (Pn) + |H' I = Yh (Pn) + nt
Yh[G(n, r, t)] > Yh (Pn) + |H' | = Yh (Pn) + nt
3
r
r
Therefore,
Yh[G(n, r, t)] = Yh(Pn) + |H'| = Yh(Pn) + nt Case 2: r = 1, 2, 3 (mod 6). Here, any Yh- set contains the set
r
L3J
H' — {v(n-2)j, V(n-3)j, V(n-8)j, V(n-9)j • • • V2j , , 1 < j < t} and so H' U V(Pn) forms a Yh-set of cardinality n + |H
Case 3: r = 4, 5 (mod 6). In this case, vertices in attached t copies are sufficient for a Yh-set of G(n,r, t). Therefore,
Yh[G(n, r, i)] = ntih(Pr) < nt( - + 1 ]. Also, any minimal HDS of G(n, r, t) requires at least
Yh[G(n, r,t)] — ntYh(Pr) > nt
vertices. Hence,
Yh[G(n, r, t)] — ntYh(Pr) — nt
r
-3.
r
L3J r
L3J
+ 1
+ 1 -
□
Definition 5 [19]. A generalized theta graph 0[nP(m)] is a graph obtained from n-internally disjoint paths, in which each path P(m) contains m internal vertices and these paths share common end vertices u and v (see Fig. 5).
Figure 5. Generalized theta graph 0[nP(m)].
Theorem 6. Yh($[nP(m)]) = <
4 + 4 + n 4 + n
m — 6
3
- m— 6 1
CO +
_
m— 6 2
CO +
_
if m = 0(mod6); if m = 1, 3, 4, 5 (mod 6); if m = 2 (mod 6).
Proof. Let us denote 0[nP(m)] by G for convenience. Clearly, {u, v} should be included in any Yh-set and any one vertex from column C1 and Cm is enough to hop dominate the vertices in C1 and Cm. The induced subgraph (G — {u, v} U Ci U C2 U Cm-i U Cm} is a collection of n-distinct paths Pm_4. As a consequence of Lemma 1, the result follows. □
Figure 6. Triangular snake T4.
Definition 6 [5]. A triangular snake graph Tn is a graph obtained from the path Pn by replacing each edge by a cycle of length 3. For example, a triangular snake T4 is shown in Fig. 6
A double triangular snake DTn consists of two triangular snakes that have a common path. That is, a double triangular snake is obtained from a path v1, v2,..., vn by joining vi and vi+1 to a new vertex xi for i = 1,2,..., n — 1 and to a new vertex yi for i = 1,2,..., n — 1. For example, a double triangular snake DT6 is illustrated in Fig. 7.
A triple triangular snake TTn is a graph in which three triangular snakes have a common path. Similarly, a four triangular snake FTn is a graph in which four triangles share a common path.
h y2 y3 y4 y5
Figure 7. Double triangular snake DT6
Remark 1. Yh(Tn) > 3.
Theorem 7.
Yh(Tn) = Yh(DTn) = yh(TTn) = Yh(FTn) =
2 +
2 +
n — 3
2 . n- 2
2 .
if n is odd (n = 3), if n is even (n = 6).
Proof. First, we observe that any Yh"set of Tn will also be a Yh"set for DTn,TTn and FTn because they share a common path.
For n = 3 and 6, Yh(Ts) = Yh(To) = 3.
Let us label the vertices of the common path as [v\, v2 ... vn} and the remaining vertices of Tn be S = [xi,i+1}, 1 < i < (n — 1) as shown in Fig. 8.
Case 1: n is odd and n > 5. While finding any Yh-set of Tn, the vertices v2 and vn-2 are taken and the remaining vertices from the subset Sk C S where
Sk — {xi,i+i| i = 0 (mod 2), 2 < i < (n — 3)}.
Clearly, |Sk| — (n — 3)/2. Thus,
Yh(Tn) < 2 +
It is easily seen that
Yh(Tn) > 2 +
Therefore,
lh(Tn) = 2+l 2
Case 2: n is even and n — 6. Note that any common vertex say xi)i+i hop dominates the vertices vi-2 and Vj of Pn, xi-2)i-i, xi+i)i+2. Equivalently, the hop degree of any vertex is at most 4. Hence by choosing vertices from the set
Sk — {xi,i+i| i = 0 (mod 2), 2 < i < (n — 2)} C S, any Yh-set can be obtained which includes the non-hop dominated vertices v2 and vn-i too. Thus,
n — 2
n — 3 2 .
n—3 2 .
n — 3
It is observed that Therefore,
Yh(Tn) < 2 + Yh(Tn) > 2 + Yh(Tn ) = 2 +
2
n-2 2 .
n-2 2 .
□
Definition 7 [5]. An alternate triangular snake ATn is a graph obtained from the path Pn, in which every alternate edge of a path is replaced by a cycle C3. For example, an alternate double triangular snake is shown in Fig. 9.
An alternate double triangular snake AD(Tn) is obtained from two alternate triangular snakes that share a common path. For example, an alternate double triangular snake is illustrated in Fig. 10.
An alternate triple (four) triangular snakes AT(Tn)(AF(Tn)) consists of three (four) alternate triangular snakes that share a common path.
Figure 9. Alternate Triangular snake AT6.
Figure 10. Alternate Triangular snake ADT5.
Theorem 8.
lh(ATn) = Yh(AD(Tn)) = Yh (AT (Tn)) = lh(AF (Tn)) =
n
2' 1 n + 1
n — 1
2 :
if n = 0, 2 (mod 4); if n = 3 (mod 4); if n = 1 (mod 4).
Proof. Let us follow the labeling of vertices as described in Theorem 7. Here, d(vi) = 3, i = 1,n and any vertex of path Pn hop dominates at most 3 vertices. In any Yh-set, it is clear that central vertices of Pn alone appear consecutively (see Fig. 11-12).
Case 1: n is even.
Case 1.1: n = 0 (mod 4). Here, any Yh-set is of the form
S = {vi| i = 2 (mod 4), 2 < i < (n — 2)} U {vj| j = 3 (mod 4), 3 < j < (n — 1)}.
Thus, Yh(ATn) < n/2. It is easily seen that, Yh(ATn) > n/2. Therefore, Yh(ATn) = n/2.
Case 1.2: n = 2 (mod 4) In this case, vn-2 must be chosen in any Yh-set and the remaining vertices are chosen from {v2,v3,v6,v7,.. .vn-4,vn-3}. Thus, Yh(ATn) < (n — 2)/2 + 1 = n/2. It is easily seen that Yh(ATn) > n/2. Therefore, Yh(ATn) = n/2.
Figure 11. ATn, when n is even.
Figure 12. ATn, when n is odd.
Case 2: n is odd.
Case 2.1: n = 1 (mod 4). In this case, any Yh-set are chosen from {v2, v3, v6, v7, • • • vn-3, vn-2}, with cardinality (n —1)/2. Thus, Yh(ATn) < (n — 1)/2 and it is easy to verify that Yh(ATn) > (n — 1)/2. Therefore, Yh(ATn) — (n — 1)/2.
Similarly, the case for n = 3 (mod 4) follows.
□
Definition 8. A Quadrilateral snake Qn is a graph obtained by replacing each edge of a path Pn by a cycle of length 4.
An alternate quadrilateral snake AQn is obtained from the path Pn by replacing its alternate edges with C4.
Proposition 1. (i) Yh(Qn) = <
( n + 2
2 ' 11 + 3
2 ' n + 1
if n = 0, 2 (mod 4); if n = 1 (mod 4); if n = 3 (mod 4).
(ii) Yh(AQn) =
11 + 1 2
n + 2 2
, if n = 1, 3 (mod 4); , if n = 0, 2 (mod 4).
Definition 9. Ciliate is a graph C(p, s) obtained from p disjoint copies of the path Ps by linking one end point of each such copy in the cycle Cp. For example, a Ciliate C(3, 3) is shown in Fig. 13.
2
• ft
Figure 13. Ciliate C(3, 3).
Remark 2. yh[C(p, q)] = p Yh(Pq)•
Definition 10. A generalized ciliate GC(p,s,t) is obtained by attaching t-copies of path Ps to each vertex of the cycle Cp.
Proposition 2. Yh[GC(3,s,t)] = <
Theorem 9. Yh[GC(4,s,t)] = <
2 + 3t
3 + 3t
3f ^ 3
3 + 3t
2 + 4t
4 + 4t s
s
-3. s
.3.
4t
3
if s = 0 (mod 6); if s = 1, 3 (mod 6);
L3 J s
-3.
+ 1, if s = 4, 5 (mod 6);
sJ
.3.
if s = 2 (mod 6).
if s = 0,1 (mod 6); if s = 2 (mod 6); if s = 3, 4, 5 (mod 6).
Proof. Let us denote the vertices in the i copy of the path Ps as {v\, v2 ...v\ : 1 < i < t}. as shown in Fig. 14. Clearly, to hop dominate the leaves and its support vertices in every ith copy of Ps, the vertices vS-2 and vS-3 (1 < i < t) have to be chosen for any Yh-set of GC(4,s,t).
Case 1: s = 0, 1 (mod 6). To hop dominate vi's and the vertices of the cycle, any Yh-set includes u2,u3. The remaining vertices in each copy of Ps in GC(4,s,t) will induce a path, thus it is sufficient to add to {vs-2,vs-3,vs-8,vs-9 ...v| ,v4}, 1 < i < s to Yh-set of GC (4, s,t). Thus,
s
Figure 14. Generalized ciliate GC(4,s,t).
Yh[GC(4,s,t)] < 2 + 4tYh(Ps) < 4t|_s/3\ and it is easily seen that Yh[GC(4,s,t)] > 2 + 4tYh(Ps)• Therefore,
Yh [GC (4,s,t)] =2 + 4tYh (Ps).
Case 2: s = 2 (mod 6). Any Yh-set comprises u1 ,u2 ,u3 ,u4 to hop dominate vl and v2 as well as the vertices of the cycle. Each copy's remaining vertices will induce a path on (s — 2) vertices. As a result, {vS—2,vS—3,vls_g,vls—9 ...v6} are required to form a Yh-set of GC(4,s,t). Thus, Yh [GC(4, s, t)] < 4 + 4t[_s/3\ and it is easy to show that Yh [GC(4, s, t)] > 4 + 4t |_s/3\. Therefore,
Yh [GC(4,s,t)] =4 + 4t|s/3\.
Case 3: s = 3, 4, 5 (mod 6). When s = 3, 5 (mod 6), H = [vis-2-3-8-
forms a Yh-set of GC(4,s,t), whereas for s = 4 (mod 6), H = {vs__ forms a Yh-set. Thus, Yh [GC (4,s,t)] = 4tYh (Ps) < 4t(|s/3\ + 1). Yh [GC(4,s,t)] > 4t(|s/3\ + 1). Therefore,
s—9
s
s—2i us—3'
32
v2}
2 u\
v\ }
It is easily seen that
Yh [GC (4,s,t)] =4t(Ls/3j +1).
□
3. Conclusion
In this study, we computed hop domination number for some special families of graphs like triangular, quadrilateral, alternate triangular, alternate quadrilateral snake graphs and examined hop domination number for some generalized graph structures like generalized theta graph, glued path graph. In future, the result obtained for generalized ciliates p = 3,4 may be extended to p > 4.
Acknowledgements
The authors thank the referees for their insightful comments. Also, the authors thank the Department of Science and Technology, Government of India for the financial support to the Department of Mathematics, SASTRA Deemed to be University under FIST Programme — Grant No.: SR/FST/MSI-107/2015(c).
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