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On the Uphill Domination Polynomial of Graphs Journal of Applied Mathematics and Physics, 2020, 8, 1168-1179 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 On the Uphill Domination Polynomial of Graphs Thekra Alsalomy, Anwar Saleh, Najat Muthana, Wafa Al Shammakh Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia How to cite this paper: Alsalomy, T., Abstract Saleh, A., Muthana, N. and Al Shammakh, W. (2020) On the Uphill Domination Po- A path π = [vv12,,, vk ] in a graph G= ( VE, ) is an uphill path if lynomial of Graphs. Journal of Applied deg v≤ deg v 1 ≤≤ik S⊆ VG Mathematics and Physics, 8, 1168-1179. ( ii) ( +1 ) for every . A subset ( ) is an uphill https://doi.org/10.4236/jamp.2020.86088 dominating set if every vertex vi ∈ VG( ) lies on an uphill path originating Received: April 20, 2019 from some vertex in S. The uphill domination number of G is denoted by Accepted: June 20, 2020 γ up (G) and is the minimum cardinality of the uphill dominating set of G. In Published: June 23, 2020 this paper, we introduce the uphill domination polynomial of a graph G. The Copyright © 2020 by author(s) and uphill domination polynomial of a graph G of n vertices is the polynomial Scientific Research Publishing Inc. = n i UP( G,, x) ∑iG=γ ( ) up( G i) x , where up( G, i) is the number of uphill do- This work is licensed under the Creative up Commons Attribution International minating sets of size i in G, and γ up (G) is the uphill domination number of G, License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ we compute the uphill domination polynomial and its roots for some families of Open Access standard graphs. Also, UP( G, x) for some graph operations is obtained. Keywords Domination, Uphill Domination, Uphill Domination Polynomial 1. Introduction In this paper, we are concerned with simple graphs which are finite, undirected with no loops nor multiple edges. Throughout this paper, we let VG( ) = n and EG( ) = m. In a graph G= ( VE, ) , the degree of v∈ VG( ) denoted by deg( v) is the number of edges that incident with v. A path in G is an alternat- ing sequence of distinct vertices. A path is an uphill path if for every 1 ≤≤ik we have deg( vii) ≤ deg( v +1 ) [1]. S nk=22 + The bistar graph kk11, with 1 vertices is obtained by joining the S non-pendant vertices of two copies of star graph k1 by new edge. The corona of two graphs G1 and G2 with n1 and n2 vertices, respectively, denoted by DOI: 10.4236/jamp.2020.86088 Jun. 23, 2020 1168 Journal of Applied Mathematics and Physics T. Alsalomy et al. GGG= 12 is obtained by taking one copy of G1 and n1 copies of G2 and joining the ith vertex of G1 with an edge to every vertex in the ith copy of G2 . The corona GK 1 (in particular) is the graph constructed by a copy of G, where for each vertex v∈ VG( ) a new vertex v′ and a pendant edge vv′ are added. The tadpole graph Tsk, is a graph consisting of a cycle graph Cs on at least three vertices and a path graph Pk on k vertices connected with bridge. The wheel graph Ws is a graph formed by connecting a single vertex to all ver- tices of a cycle graph Cs . The book graph is a Cartesian product Bmm= SP × 2 , where Sm is the star graph with m +1 vertices and P2 is the path graph on two vertices. Also, the windmill graph Wd( s, k ) is a graph constructed for s ≥ 2 and k ≥ 2 by joining k copies of the complete graph Ks at a shared universal vertex. The dutch windmill graph D( sk, ) is the graph obtained by taking k copies of the cycle graph Cs with a vertex in common. Also, the friendship Fk is a graph that constructed by joining k copies of the cycle graph C3 and observes that Fk is a special case of D( sk, ) . Finlay, the firefly graph Fstk,, with stk,,≥ 0 and n=22 s + tk ++ 1 vertices is defined by consisting of s triangles, t pendent paths of length 2 and k pendent edges, sharing a com- mon vertex. Any terminology not mentioed here we refer the reader to [2]. A set SV⊆ of vertices in a graph G is called a dominating set if every ver- tex vV∈ is either vS∈ or v is adjacent to an element of S, The uphill do- minating set “UDS” is a set SV⊆ having the property that every vertex vV∈ lies on an uphill path originating from some vertex in S. The uphill do- mination number of a graph G is denoted by γ up (G) and is defined to be the minimum cardinality of the UDS of G. Moreover, it’s customary to denote the UDS having the minimum cardinality by γ up (G) -set, for more details in domi- nation see [3] and [4]. Representing a graph by using a polynomial is one of the algebraic representa- tions of a graph to study some of algebraic properties and graph’s structure. In general graph polynomials are a well-developed area which is very useful for analyzing properties of the graphs. The domination polynomial [5] and the uphill domination of a graph [6], mo- tivated us to introduce and study the uphill domination polynomial and the uphill domination roots of a graph. 2. Uphill Domination Polynomial Definition 2.1. For any graph G of n vertices, the uphill domination poly- nomial of G is defined by n UP(,) G x= ∑ up( G,, i) xi iG=γup ( ) where up( G, i) is the number of uphill dominating sets of size i in G. The set of roots of UP( G, x) is called uphill domination roots of graph G and denoted by ZGup ( ) . Example 2.2. The uphill domination polynomial of House graph H (as shown DOI: 10.4236/jamp.2020.86088 1169 Journal of Applied Mathematics and Physics T. Alsalomy et al. in Figure 1) with 6 vertices and γ up (H ) = 2 is given by 2 3 4 56 UP( H, x) = 2795 x ++++ x x x x . Furthermore, ZHup ( ) ={0, −− 1, 2} . The following theorem gives the sufficient condition for the uphill domina- tion polynomial of r-regular graph. Theorem 2.3. Let G be connected graph with n ≥ 2 vertices. Then, n UP( G,1 x) =+−( x) 1 if and only if G is r-regular graph. Proof. Let G be a connected graph of n ≥ 2 vertices. Suppose that the uphill domination polynomial of G is given by n n 2 n UP( G,1 x) =( + x) −= 1 nx + x + + x . 2 Since the first coefficient of the polynomial is n, then it is easily verified that for every v∈ VG( ) , the singleton vertex set {v} is an UDS in G. Assume that G is not r-regular graph. Hence there exists a vertex u∈ VG( ) such that deg( u) = s ≠ r . Now, we have two cases: Case 1: If sr> , then the set {u} is not UDS which contradict that every singleton vertex set is an UDS in G. Case 2: If sr< , then for all uv≠ with deg( v) = r , we get the set {v} is not UDS which is also contradict that every singleton vertex set is an UDS in G. Thus, G must be r-regular graph. On the other hand, suppose that G is r-regular graph with n ≥ 2 vertices. We have γ up (G) = 1 , then there exist n UDS of size one, while for i = 2 there are n UDS and so on. Thus, we can write the uphill domination polynomial as 2 nn23 nn n UP( G, x) = nx + x + x ++ x =+(11 x) −. 22 n Corollary 2.4. Let G ba a graph with s vertices. If G is a cycle Cs or com- s plete graph Ks , then UP( G,1 x) =+−( x) 1. Corollary 2.5. The uphill domination polynomial for the regular graph sk GCC=sk × with sk vertices is given by UP( G,1 x) =+−( x) 1. Corollary 2.6. [6] Let G be a graph with m components. Then, m γγup (GG) = ∑ up( i ). j=1 Proposition 2.7. If a graph G with n vertices consists of m components GG12, ,, Gm , then Figure 1. The House graph. DOI: 10.4236/jamp.2020.86088 1170 Journal of Applied Mathematics and Physics T. Alsalomy et al. m UP( G, x) = ∏ UP( Gi ,. x) i=1 Proof. By using mathematical induction we found that for m = 1 the state- ment is true and the proof is trivial. Suppose that the statement is true when mk= such that k UP( G, x) = ∏ UP( Gi ,. x) i=1 Now, we prove that the statement is true when mk= +1. Let G consists of k +1 components that mean GG= 12 G Gk + 1. If the set {rr12,, , rk + 1} represent the uphill domination number for the components of G respectively, such that γ up(Gr i) = i ∀≤≤11ik +. Then, by Corollary (2.6) it easily to see that γγ= = γ =++ = up (G) up Gi ∑ up( Gr i ) 11 rk + r. 11≤≤ik + 11≤≤ik + Thus, up( G, r) is exactly equal the number of way for choosing an UDS of size r1 in G1 and an UDS of size r2 in G2 and so on.
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