Effects of Some Operations on Domination Chromatic Number in Graphs
EEffects of Some Operations on Domination ChromaticNumber in Graphs
Yangyang Zhou, Dongyang Zhao
Peking University, Beijing, China
Abstract
For a simple graph G , a domination coloring of G is a proper vertex coloringsuch that every vertex of G dominates at least one color class, and every colorclass is dominated by at least one vertex. The domination chromatic num-ber, denoted by χ dd ( G ), is minimum number of colors among all dominationcolorings of G . In this paper, we discuss the effects of some typical oper-ations on χ dd ( G ), such as vertex (edge) removal, vertex (edge) contraction,edge subdivision, and cycle extending. Keywords:
Domination chromatic number, Removal, Contraction, Edgesubdivision, Cycle extending
1. Introduction
Let G = ( V, E ) be a simple graph. For any vertex v ∈ V ( G ), the openneighborhood of v is the set N ( v ) = { u | uv ∈ E ( G ) } and the closed neighbor-hood is the set N [ v ] = N ( v ) ∪ { v } . The degree of a vertex v ∈ V , denotedby deg ( v ), is the cardinality of its open neighborhood. The maximum andminimum degree of a graph G is denoted by ∆( G ) and δ ( G ), respectively.For a subset X ⊆ V , we denote by G [ X ] the subgraph of G induced by X . We consider only finite, undirected and simple connected graphs in thispaper. The reader is referred to the book of Bondy and Murty [1] for anyundefined terms.A proper vertex k -coloring of a graph G = ( V, E ) is a mapping f : V →{ , , · · · , k } such that any two adjacent vertices receive different colors. In Email address: [email protected] (Yangyang Zhou)
Preprint submitted to Journal Name September 13, 2019 a r X i v : . [ c s . D M ] S e p act, this problem is equivalent to the problem of partitioning the vertex setof G into k independent sets { V , V , · · · , V k } where V i = { x ∈ V | f ( x ) = i } .The set of all vertices colored with the same color is called a color class. Thechromatic number of G , denoted by χ ( G ), is the minimum number of colorsneeded in a proper coloring of G .A dominating set S is a subset of the vertices in G such that every vertexin G either belongs to S or has a neighbor in S . The domination number γ ( G ) is the minimum cardinality of a dominating set of G .Coloring and domination are two important fields in graph theory, andboth of them have rich research results. For comprehensive surveys of color-ing and domination in graphs, refer [2, 3, 4, 5] and [6, 7, 8, 9], respectively.Also, relations between the two fields have been discussed in many ways.Chellali and Volkmann [10] showed some relations between the chromaticnumber and some domination parameters in a graph. Hedetniemi et al. [11]introduced the concept of dominator partition of a graph. Motivated by [11],Gera et al. [12] proposed the dominator coloring as a proper coloring suchthat every vertex has to dominate at least one color class (possibly its ownclass) in 2006. Gera researched further on this coloring problem in [13, 14].Similar to the dominator coloring, Kazemi [15] studied the concept of a totaldominator coloring in 2015, which is a proper coloring such that each vertexof the graph is adjacent to every vertex of some (other) color class. Moreresults on the dominator coloring could be found in [16, 17, 18, 19]. Then,in 2015, Boumediene et al. [20] proposed the dominated coloring as a propercoloring where every color class is dominated by at least one vertex. Basedon these studies, we introduce the domination coloring in [21], in which boththe vertices and color classes should satisfy the domination property.A domination coloring of G is a proper vertex coloring such that everyvertex of G dominates at least one color class (possibly its own class), andevery color class is dominated by at least one vertex. The domination chro-matic number χ dd ( G ) is the minimum number of color classes in a dominationcoloring of G . In [21], we give basic properties of χ dd ( G ), and prove that the k -domination coloring problem is NP-Complete for k ≥ G ? On this problem, thedomination-critical concepts of graphs are proposed to examine the effects ofvertex (edge) removal and addition on γ , more results refer [22, 23, 24, 25, 26].Ghanbari and Alikhani [27] study the effects of some operations on the to-2al dominator chromatic number in 2016. In 2018, Alikhani, Ghanbari andSoltani [28] discuss the total dominator chromatic number of k -subdivisionof a graph. In this paper, we aim to study the effects on χ dd ( G ) when weconsider some operations on a graph G . In Section 2 and Section 3, we con-sider the basic removal and contraction operations on the vertex and edge.In Section 4, we examine the effects of edge subdivision. In Section 5, westudy the cycle extending operation and its effects on χ dd ( G ). In Section 6,we make conclusion and propose the future research.
2. Vertex and edge removal
This section focuses on the influence of vertex and edge removal on thedomination chromatic number. Let v be a vertex and e be an edge of G ,respectively. The graph G − v is obtained by deleting the vertex v and alledges incident with v in G . And G − e is a graph that obtained from G bysimply removing the edge e . In the following, we discuss the effect on χ dd ( G )when G is changed by the vertex and edge removal. Theorem 1.
Let G be a connected graph, and v ∈ V ( G ) is not a cut vertex,then χ dd ( G ) − ≤ χ dd ( G − v ) ≤ χ dd ( G ) + deg ( v ) − . Proof. We first prove the left inequality. Given any domination coloringof G − v , we consider the domination coloring of G as follows. If we addvertex v and all the corresponding edges to G − v , then it suffices to adda new color i to v and all other colors unchanged. Since v dominates thecolor class itself and every vertex except v dominates the old color classes,the color class v itself can be dominated by any adjacent vertices of v andall the other color classes be dominated by the original vertices, we obtain adomination coloring for G . So, χ dd ( G ) ≤ χ dd ( G − v ) + 1.Now we prove the right part χ dd ( G − v ) ≤ χ dd ( G ) + deg ( v ) −
1. For anydomination coloring of G , suppose that vertex v have color i , then we havetwo cases:Case 1. There exsist other vertices with color i . If there exsists a colorclasses which is only dominated by v , then we give each vertex of this colorclass a new color. Obviously, the number of these vertices is at most deg ( v ).Each of the new colored vertices dominates the color class itself, and eachof them as a color class is dominated by another neighbor except v , which3ust exsist, since v is not a cut vertex. So, we get a domination coloring of G − v , and χ dd ( G − v ) ≤ χ dd ( G ) + deg ( v ) −
1. If there exsist no color classdescribed above, then we can obtain a domination coloring of G − v usingthe old color of G . So, χ dd ( G − v ) ≤ χ dd ( G ).Case 2. There exsist no other vertices with color i . As the analysis inCase 1, we have χ dd ( G − v ) ≤ χ dd ( G ) + deg ( v ) − v , and otherwise, χ dd ( G − v ) ≤ χ dd ( G ) − χ dd ( G − v ) ≤ χ dd ( G ) + deg ( v ) −
1, and the theorem follows. (cid:3)
Theorem 2.
Let G be a connected graph, and e ∈ E ( G ) is not a cut edge,then χ dd ( G ) − ≤ χ dd ( G − e ) ≤ χ dd ( G ) + 2 . Proof. Let e = uv ∈ E . First, we prove the left inequality. We shallpresent a domination coloring for G − e . If we add the edge e to G − e ,then there exist two cases. If two vertices u and v have the same color inthe domination coloring of G − e , then we add a new color i to one of them.Since each vertex use the old color, we obtain a domination coloring for G .So we have χ dd ( G ) ≤ χ dd ( G − e ) + 1. If two vertices u and v do not have thesame color in the domination coloring of G − e , then the domination coloringof G − e can also be a domination coloring for G . So, χ dd ( G ) ≤ χ dd ( G − e ).Therefore, χ dd ( G ) − ≤ χ dd ( G − e ).Now, we prove χ dd ( G − e ) ≤ χ dd ( G ) + 2. For a domination coloring of G , suppose that u ∈ V i and v ∈ V j , that is u has color i and v has color j .Then, we have the following cases:Case 1. The vertex u does not dominate the color class V j , and the vertex v does not dominate the color class V i . In this case, the domination coloringof G gives a domination coloring of G − e . So, χ dd ( G − e ) ≤ χ dd ( G ).Case 2. The vertex u dominate the color class V j , and the vertex v doesnot dominate the color class V i . By definition, u adjacents to every vertexin V j , V i must be dominated by some other vertices. We give a new color k to vertex v in G − e . Thus, u dominate the color class formed by V j /v , v dominate the color class itself. Since e is not a cut edge, G − e is connectedand v has other neighbors. So there have other vertices dominate the colorclass v itself. Each other vertex use the old color, so we obtain a dominationcoloring of G − e , and χ dd ( G − e ) ≤ χ dd ( G ) + 1.Case 3. The vertex u dominate the color class V j , and the vertex v dominate the color class V i . In this case, we give two new colors k and l to4ertices u and v , respectively. In the new obtained coloring, both u and v dominate the color class themselves, and there exist other vertices adjacentto u and v by the connectedness of G − e , respectively. Since each vertexuse the old color, the new coloring is a domination coloring of G − e . So, χ dd ( G − e ) ≤ χ dd ( G ) + 2.By the above three cases, we have χ dd ( G − e ) ≤ χ dd ( G ) + 2. So, the resultfollows. (cid:3)
3. Vertex and edge contraction
For a simple garph G , S ⊆ V ( G ) is a subset of vertices. The contractionof S in G , denoted by G ◦ S , is the graph obtained by replacing all vertices in S with a single new vertex, such that edges insident with the new vertex arethose edges that were insident with any vertex in S . Specially, for S = { u, v } ,we consider the following two cases:(1) Vertex u and v are adjacent in G , that is e = uv ∈ E ( G ). Thecontraction of an edge e = uv in G , denoted by G ◦ e , is the graph obtainedby deleting the edge e , and replacing u and v with a new single vertex suchthat vertices adjacent to the new vertex if they were adjacent to vertex u or v , as shown in Figure 1, where the black points around the vertex denotethat there may be edges insident with the vertex.(2) Vertex u and v are not adjacent, uv / ∈ E ( G ). We denote this contrac-tion operation by G ◦ { u, v } , as shown in Figure 2. x G G e
Edge contraction eu v
Figure 1: The edge contraction operation
In this section, we discuss the effect on χ dd ( G ) when G is changed by thevertex and edge contraction. Theorem 3.
Let G be a connected graph, e ∈ E ( G ) , then χ dd ( G ) − ≤ χ dd ( G ◦ e ) ≤ χ dd ( G ) + 1 . { , } G u v
Vertex contraction u v x w Figure 2: The vertex contraction operation
Proof. Let e = uv ∈ E . We first prove the right inequality. Let f be a χ dd ( G )-domination coloring of G , f ( u ) = i and f ( v ) = j . Now we present adomination coloring for G ◦ e . We denote x the replacement vertex of u and v .For G ◦ e , we give a new color k to vertex x and each other vertices remain theprevious coloring in f . The obtained coloring is a domination coloring of G ◦ e ,since x dominate the color class itself and this color class must be dominatedby any adjacent vertices of x , and all other vertices and color classes remainthe old domination relationship. So, we have χ dd ( G ◦ e ) ≤ χ dd ( G ) + 1.Now, we prove the left inequality. Consider any χ dd ( G ◦ e )-dominationcoloring of G ◦ e . By adding the removed vertex and all the correspondingedges to G ◦ e , we get the graph G . Then remove the old color of the endpointsof e , add two new colors i and j to the vertices u and v , respectively, andkeep the old colors of all other vertices. We get a new coloring for G , and itis easy to check this is a domination coloring. So, χ dd ( G ) ≤ χ dd ( G ◦ e ) + 2.Thus, the result follows. (cid:3) Theorem 4.
For a connected graph G , we have χ dd ( G ) − ≤ χ dd ( G ◦ { u, v } ) ≤ χ dd ( G ) + 1 . Proof. Let u, v are two unadjacent vertices of G . We first prove theright inequality. For a χ dd ( G )-domination coloring f of G , we will present adomination coloring of G ◦{ u, v } based on f . We give a new color k to the newvertex, denoted by x , and all the other vertices remain the previous coloringin f . Since each vertex other than x satisfies the domination properties, and x dominate the color class itself or the one which is dominated by u or v in f ,also the color class { x } must be dominated by any adjacent vertices of x , weobtain a domination coloring of G ◦ { u, v } . So, χ dd ( G ◦ { u, v } ) ≤ χ dd ( G ) + 1.For the left part, we first present a χ dd ( G ◦ { u, v } )-domination coloringof G ◦ { u, v } . Then, we add the removed vertex and the corresponding edges6o G ◦ { u, v } , in order to get G . In G , remove the old coloring of u and v and give two new colors to them, and keep other vertices colors unchanged.Like before, we can check this is a domination coloring of G . So, we have χ dd ( G ) ≤ χ dd ( G ◦ { u, v } ) + 2.The result follows. (cid:3)
4. Edge subdivision
Edge subdivision is a typical operation in graphs. Let G be a connectedgraph and k be a positive integer. The k -subdivision of G , denoted by S k ( G ),is the graph obtained from G by replacing each edge with a path of length k . For each edge e = v i v j ∈ E ( G ), let P v i ,v j denote the k -path that replacing v i v j , and we call P v i v j a superedge, any new vertex of P v i v j is an internalvertex. Note that if k = 1, then S ( G ) = G .In this section, we investigate the domination chromatic number of the k -subdivision of graph G , and the relation between χ dd ( G ) and χ dd ( S k ( G )). Theorem 5.
Let G be a connected graph with m edges, and k ≥ , then χ dd ( P k +1 ) ≤ χ dd ( S k ( G )) ≤ ( m − χ dd ( P k ) + χ dd ( P k +1 ) . Proof. We first prove the left inequality. Let e = uv be an arbitrary edgeof G , and P uv = ux x · · · x k − v be the replacement of uv in S k ( G ), as shownin Figure 3. We shall get a domination coloring of P k +1 base on a dominationcoloring of S k ( G ). Let f a χ dd -domination coloring of S k ( G ). We considerthe restriction of f to the induced subgraph P uv and denote it by f (cid:48) . In f ,both of the endpoints u and v dominate either the color class itself or thecolor class formed by some adjacent vertices of them, respectively. Thus,they also dominate the color class itself or the color class formed by someadjacent vertices of them in f (cid:48) . With the internal vertices and their coloringsunchanged, we conclude that each color class of f (cid:48) must be dominated bythe vertex itself or the same vertex as in f . So, f (cid:48) is a domination coloringof P uv , and we have χ dd ( P k +1 ) ≤ | f (cid:48) | ≤ | f | = χ dd ( S k ( G )).Now, we prove the right inequality. Let G be a connected graph with m edges. For any vertex u ∈ V ( G ), let N G ( u ) = { v , v , · · · , v p } . Theedges uv i are replaced with the superedges P uv i ( i = 1 , , · · · , p ), respectively.We can easily color the vertices of superedge P uv with χ dd ( P k +1 ) colors asa domination coloring of P k +1 . Then, we color the vertices of P uv i , i =7 ( ) k S G
Edge subdivision eu v u v x x k x Figure 3: The edge subdivision operation , , · · · , p , such that vertex u has the unique color in P uv i ( i = 2 , · · · , p ) asthe color it has in P uv , and the following formula is satisfied: { f ( P uv i ) ∩ f ( P uv j ) }\{ f ( u ) } = ∅ , where i, j = 1 , , · · · , p and i (cid:54) = j , f ( P uv i ) is the set of colors of vertices in P uv i . Thus, we obtain a domination coloring of P uv i ( i = 1 , , · · · , p ) with atmost ( p − χ dd ( P k ) + χ dd ( P k +1 ) colors. Next, we consider the superedgeswhich are uncolored and are replacements of edges incident with vertices v i , i = 1 , , · · · , p . Continue the above process until all vertices of S k ( G )are colored. Note that some vertices will be recolored when they lie on somecycles. The coloring obtained finally is a domination coloring of S k ( G ), whichuse at most ( m − χ dd ( P k ) + χ dd ( P k +1 ) colors. So, we get the right part andthe theorem follows. (cid:3)
5. Cycle extending
In thia section, we focus on the cycle extending operation in graphs, andstudy its influence on the domination chromatic number. For a connectedgraph G , C is a cycle of length l ( ≥
3) in G . The cycle extending based on C is adding a new vertex x and connecting x to every vertex on C . Wedenote the obtained graph by W + G ( G ), which is shorted for W + ( G ) withoutconfusion. The operation is shown in Figure 4, where the black points aroundthe vertices denote that there may be edges, and the black points betweenedges denote that the length of cycle C is at least 3. Theorem 6.
Let G be a connected graph and C be any cycle of length l in G , then χ dd ( G ) − l ≤ χ dd ( W + ( G )) ≤ χ dd ( G ) + 1 . Proof. We first consider the right part of the inequality. Let f be adomination coloring of G . We can get a domination coloring of W + ( G )8 G ( ) W G Cycle extending
Figure 4: The cycle extending operation based on f by adding a new color to the center vertex x and remaining othercolors of vertices unchanged. Thus, χ dd ( W + ( G )) ≤ χ dd ( G ) + 1.Now, we prove the left part. Let f ∗ be a domination coloring of W + ( G ).We aim to get a domination coloring of graph G based on f ∗ . There exsisttwo cases for the color of the center vertex x in f ∗ .Case 1. x has the unique color in f ∗ , that is, x itself is a color class.In this case, we need to consider the vertices which only dominate the colorclass formed by x and the color classes which are only dominated by x , andall these elements are vertices on cycle C . So, just adding at most l newcoloring to these vertices, we can obtain a domination coloring of G . Theinequality follows.Case 2. There exsist some other vertices having the same color as x . Wedenote the color class containing x by V i . Since the domination propertiesof V i and all other vertices are not affected by the deletion of x , we need toconsider the color classes that are dominated by x only. Also, all these colorclasses must be formed by vertices on cycle C . Thus, we get a dominationcoloring of G by adding at most l colors like Case 1. The left inequality isobtained.So, the theorem follows. (cid:3)
6. Conclusion and Further Research
In this paper, we study the effects on the domination chromatic number χ dd ( G ) of a graph G when G is modified by some operations related tothe vertices, edges and cycles of G . Specificly, several general bounds andproperties of the new created graphs are given with respect to χ dd ( G ).For the future research, we plan to examine the effects of some complexoperations on the domination chromatic number, such as the edge filping9peration, the complementary operation and some binary operations. Also,we are interest in the operations which do not affect the original dominationchromatic number. Acknowledgements
The authors acknowledge support from Peking University. The authorsalso would like to thank referees for their useful suggestions.
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