Graph burning is a discrete-time process on graphs [1], Bonato et al. introduced the concept of the burning number to measures the speed of contagion spread on a graph and denoted the burning number of graph G by b ( G ) in [2], some special classes of graphs has been studied, such as spider graphs [3], path forest [4], generalized Petersen graphs [5], theta graphs [6], caterpillars [7] and fence graphs [8]. For a survey on graph burning see [9]. Later, Li et al. [10] generalized the burning number and introduced the generalized burning number b r ( G ) of G for r ≥ 1 . Follow these, Song et al. [11] propose the Independent Cascade graph burning model of G , where a burned vertex v can burn its neighbor w only if the influence that v exerts on w is larger than a given threshold β . Note that when β = 0 , it is a traditional burning problem. The task is still to find the minimum sequence of vertices that can burn the whole graph. The minimum number of time steps is IC burning number b β ( G ) of graph G . In the burn process of IC model of graph G , we call a vertex is fire source, it is selected to burn. For a given threshold β , the x i is the i -th fire source in the β -burning process of G . If G are burned after k time steps, we call the fire source sequence ( x 1 , x 2 , ⋯ , x k ) a β -burning sequence of G . Clearly, the IC burning number b β ( G ) is the minimum length among all β -burning sequences of graph G .

In reality, the influence u receives from its neighbor w is an arbitrary value in [ 0 , 1 ] , we easily know whether a vertex can be burned by its neighbor depends heavily on its degree. For v , u ∈ V ( G ) , the distance between them is denoted by d ( u , v ) . The open neighborhood N ( v ) is the set of vertices at distance one from a vertex v . Clearly, the closed neighborhood N [ v ] = N ( v ) ∪ { v } . Given a positive integer k and fraction β ∈ [ 0 , 1 ] , the k -th closed β -neighborhood of u is a set { v ∈ V ( G ) : d ( u , v ) ≤ k , f ( u ) ≥ β } and is denoted by N β k [ v ] , where f ( u ) = 1 d ( u ) .

For h ≥ 1 , k ≥ 2 , a perfect k -ary tree with height h , denoted T k h , is a tree with k h leaves and a root vertex with degree k whose distance to all the leaves is h and all other internal vertices have degree k + 1 . The height of a vertex is the number of edges present in the path connecting that vertex to a leaf vertex. We call the internal vertices that are the parents of leaves as parent-leaves.

A spider is a tree contain one vertex called the spider head with degree at least 3. In a spider graph, every leaf is connected to the head by a path which called an arm. we denote such a spider graph by S P ( s , r ) if all the arms of the spider graph with maximum degree s are of the same length r .

A ( n , 1 ) -lollipop graph L n , 1 is a graph with V ( L n , 1 ) = { v , u 1 , u 2 , ⋯ , u n } and E ( L n , 1 ) = { u i u j | i , j = 1 , 2 , ⋯ , n , i ≠ j } ∪ { v u 1 } , see Figure 1(a). A corona graph of K n and K 1 , denoted by K n ∘ K 1 , is graph with V ( K n ∘ K 1 ) = { v 1 , v 2 , ⋯ , v n , u 1 , u 2 , ⋯ , u n } and E ( K n ∘ K 1 ) = { u i u j | i , j = 1 , 2 , ⋯ , n , i ≠ j } ∪ { v i u i | i = 1 , 2 , ⋯ , n } , see Figure 1(b).

(a) (b)

Figure 1. ( n , 1 ) -lollipop graph L n , 1 and corona graph K n ∘ K 1 .

All graphs considered in this paper are finite and simple. We use book [12] for notation and terminology not defined here. In this paper, we study the β -burning problem on several graph including complete k -ary tree, spider graphs, ( n , 1 ) -lollipop graph, corona graph of K n and K 1 and other graphs such as sunflower graph, friendship graph and Dutch windmill graph.

Proposition2.1. [11] G isaconnectedgraphwith n verticesand f ( v i ) < β for 1 ≤ i ≤ k , where 0 < β ≤ 1 .Then, k ≤ b β ( G ) ≤ n .

Proposition2.2. [11] G isagraphwith n verticesand Δ ( G ) isthemaximumdegree.Then, ( x 1 , x 2 ) isanoptimum β -burningsequencefor G ifandonlyifoneofthefollowingconditionsismet:

1) Δ ( G ) = d ( x 1 ) = n − 1 , and f ( w ) ≥ β for all ∀ w ∈ N ( x 1 ) \ { x 2 } .

2) Δ ( G ) ≥ d ( x 1 ) = n − 2 , and f ( w ) ≥ β for all ∀ w ∈ N ( x 1 ) .

In [13], Bonato et al. provide a number of properties of the burning number.

Proposition2.3. [13]If T isatreeand H isasubtreeof T , thenwehavethat b ( H ) ≤ b ( T ) .

A subgraph H of graph G is called an isometric subgraph if we have d H ( u , v ) = d G ( u , v ) .

Proposition2.4. [13]Let H beanisometricsubgraphofagraph G andforanynode x ∈ V ( G ) \ V ( H ) , andanypositiveinteger k , thereexistanode f k ( x ) ∈ V ( H ) satisfies N k [ x ] ∩ V ( H ) ⊆ N k H [ f k ( x ) ] .Thenwehavethat b ( H ) ≤ b ( G ) .

Proposition2.5. [13]Foranygraph G withradius r anddiameter d , wehavethat

Proposition2.6. [13]Foragraph G , wehavethat

b ( G ) = min { b ( T ) : T is a spanning subtree of G } .

In this section, we determined the IC burning number of some operation graphs. First, we consider the perfect binary tree, spider graphs, ( n , 1 ) -lollipop graph, corona graph K n ∘ K 1 .

Theorem3.1.Let T 2 h beaperfectbinarytreeofheight h , where h ≥ 2 .Then

Proof. Let s be the root of T 2 h and the leaves of T 2 h be u i for 1 ≤ i ≤ 2 h . Suppose the parent-leaves of T 2 h be v i for 1 ≤ i ≤ 2 h − 1 .

Case1: 0 ≤ β ≤ 1 3

The T 2 h has total h + 1 levels, since T 2 h has height h . Suppose we let x 1 = s . At the t -th step ( t = h + 1 ) , the fire completes the burning of the t -th level of T 2 h . Therefore, b β ( T 2 h ) ≤ h + 1 . Now, suppose S = ( x 1 , x 2 , ⋯ , x k ) is a β -burning sequence of T 2 h , where k < h + 1 . The fire source x i can spread to the vertices of N k − i [ x i ] . For 1 ≤ i ≤ k , N k − i [ x i ] can contain at most 2 k − i leaf vertices of T 2 h . Therefore, we burned at most ∑ i = 1 k   2 k − i = 2 k − 1 leaf vertices. Since the total number of leaves of T 2 h is 2 h . Now, 2 h ≥ 2 k > 2 k − 1 , a contradiction, we have b β ( T 2 h ) ≥ h + 1 . Thus b β ( T 2 h ) = h + 1 .

Case2: 1 3 < β ≤ 1 2

Let two children of s be s 1 and s 2 . Suppose S = { s , u 1 , u 2 , ⋯ , u 2 h } , Y = { s 1 , s 2 , v 1 , v 2 , ⋯ , v 2 h − 1 } and X = S ∪ Y . The case for h = 2 , by simple checking, we know that b β ( T 2 h ) = 3 . Next, the case h ≥ 3 . Since 1 3 < β ≤ 1 2 , thus T 2 h \ S must be fire sources. Let x 1 = s 1 , x 2 = s 2 and x i = v i − 2 for 3 ≤ i ≤ 2 h − 1 + 2 . Further, we choose T 2 h \ X as remain fire sources. Obviously, ( x 1 , x 2 , ⋯ , x 2 h − 2 ) is a β -burning sequence of T 2 h and thus b β ( T 2 h ) ≤ 2 h − 2 . Now, we show b β ( T 2 h ) ≥ 2 h − 2 . As f ( V ( T 2 h ) \ S ) < β for 1 ≤ i ≤ 2 h − 2 . By Proposition 2.1, we get 2 h − 2 ≤ b β ( T 2 h ) . Then, b β ( T 2 h ) = 2 h − 2 .

Case3: 1 2 < β ≤ 1

Let U = { u i | 1 ≤ i ≤ 2 h } , V = { v i | 1 ≤ i ≤ 2 h − 1 } and H = U ∪ V . We choose x i = v i for 3 ≤ i ≤ 2 h − 1 and V ( T 2 h ) \ H as remain fire sources. Thus, b β ( T 2 h ) ≤ 2 h − 1 . On the other hand, f ( V ( T 2 h ) \ U ) < β for 1 ≤ i ≤ 2 h − 1 , combine Proposition 2.1, we have 2 h − 1 ≤ b β ( T 2 h ) . Therefore, b β ( T 2 h ) = 2 h − 1 .

Corollary3.2.Let T k h beaperfect k -arytreeofheight h andorder n , where h ≥ 3 and k ≥ 3 .Then

Proof. Let s be the root of T k h and the leaves of T k h be u i for 1 ≤ i ≤ k h . Suppose the parent-leaves of T k h be v i for 1 ≤ i ≤ k h − 1 .

For the case 0 ≤ β ≤ 1 k + 1 , consider T 2 h is a subtree of T k h . Combine Proposition 2.3 and Theorem 3.1, we directly get b β ( T k h ) ≥ b β ( T 2 h ) = h + 1 . On the other hand, the fact that b β ( T k h ) ≤ h + 1 can be determined by let x 1 = s . Then, b β ( T k h ) = h + 1 .

The case for 1 k + 1 < β ≤ 1 k , let the children of s be s i for 1 ≤ i ≤ k . Suppose S = { u i | 1 ≤ i ≤ k h } ∪ { s } , Y = { s i | 1 ≤ i ≤ k } ∪ { v j | 1 ≤ j ≤ k h − 1 } and X = S ∪ Y . Similarly with Theorem 3.1, we have that b β ( T k h ) ≤ n − k h − 1 . Further, f ( V ( T k h ) \ S ) < β , by Proposition 2.1, we directly get n − k h − 1 ≤ b β ( T k h ) . Then, b β ( T k h ) = n − k h − 1 .

Consider that 1 k < β ≤ 1 , let U = { u i | 1 ≤ i ≤ k h } , V = { v i | 1 ≤ i ≤ k h − 1 } and H = U ∪ V . Similarly, we choose V ( T k h ) \ U be fire sources. Clearly, ( x 1 , x 2 , ⋯ , x n − k h ) is a β -burning sequence of T k h . So, b β ( T k h ) ≤ n − k h . On the other hand, since f ( V ( T k h ) \ U ) < β , combine Proposition 2.1, we have that n − k h ≤ b β ( T k h ) . Therefore, b β ( T k h ) = n − k h .

Theorem3.3.Let S P ( s , r ) beaspidergraphoforder n for s ≥ r .Then

Proof. Let leaf vertices of S P ( s , r ) be Y = { v i | 1 ≤ i ≤ s } . Suppose the neighbour vertices of leaf vertices are U = { u i | 1 ≤ i ≤ s } and X = Y ∪ U .

The case for 0 ≤ β ≤ 1 2 , we first discuss S P ( r , r ) . The radius of S P ( r , r ) is r . So by Proposition 2.5, we have that b β ( S P ( r , r ) ) ≤ r + 1 . Now we show that b β ( S P ( r , r ) ) ≥ r + 1 . Assume b β ( S P ( r , r ) ) ≤ r and S = ( x 1 , x 2 , ⋯ , x t ) is a β -burning sequence of S P ( r , r ) , where t ≤ r . No fire source in S shall be able to burn more than 1 leaf vertex in S P ( r , r ) , since d ( v i , v j ) = 2 r for i , j = 1 , 2 , ⋯ , s and i ≠ j . Thus S will be able to burn at most t leaf vertices. This implies that at least 1 leaf vertex will be left unburned, a contradiction, thus we have that b β ( S P ( r , r ) ) ≥ r + 1 . Therefore, b β ( S P ( r , r ) ) = r + 1 . As S P ( r , r ) is a subtree of S P ( s , r ) , by Proposition 2.4 and Proposition 2.5, we have b β ( S P ( s , r ) ) ≤ r + 1 .

For the case 1 2 < β ≤ 1 . Consider that f ( V ( S P ( s , r ) ) \ Y ) < β , by Proposition 2.1, we have that n − s ≤ b β ( S P ( s , r ) ) . Now, we show that b β ( S P ( s , r ) ) ≤ n − s . Let x i = u i for 1 ≤ i ≤ s and V ( S P ( s , r ) ) \ X be remain fire sources. Obviously, ( x 1 , x 2 , ⋯ , x n − s ) is a β -burning sequence of S P ( s , r ) and then b β ( S P ( s , r ) ) ≤ n − s . Further, we have that b β ( S P ( s , r ) ) = n − s for 1 2 < β ≤ 1 .

Theorem3.4.Let K n ∘ K 1 beacoronagraphof K n and K 1 withorder 2 n ( n ≥ 2 ) .Then

Proof. Let V ( K n ∘ K 1 ) = { v 1 , v 2 , ⋯ , v n , u 1 , u 2 , ⋯ , u n } and E ( K n ∘ K 1 ) = { u i u j | i , j = 1 , 2 , ⋯ , n , i ≠ j } ∪ { v i u i | i = 1 , 2 , ⋯ , n } . The case for 0 ≤ β ≤ 1 n , combine Proposition 2.2, we directly get b β ( K n ∘ K 1 ) ≥ 3 . Now, we show b β ( K n ∘ K 1 ) ≤ 3 . Let x 1 = u 1 , x 2 = v 1 and x 3 = v n . Clearly, ( x 1 , x 2 , x 3 ) is a β -burning sequence of K n ∘ K 1 and thus b β ( K n ∘ K 1 ) ≤ 3 . Therefore, b β ( K n ∘ K 1 ) = 3 . Consider the case 1 n < β ≤ 1 , let x i = u i for 1 ≤ i ≤ n . Clearly, ( x 1 , x 2 , ⋯ , x n ) is a β -burning sequence of K n ∘ K 1 . Thus b β ( K n ∘ K 1 ) ≤ n . Now, we show b β ( K n ∘ K 1 ) ≥ n . Obviously, f ( u i ) < β for 1 ≤ i ≤ n . By Proposition 2.1, we get n ≤ b β ( K n ∘ K 1 ) . Then, b β ( K n ∘ K 1 ) = n .

Corollary3.5.Let L n , 1 bea ( n , 1 ) -lollipopoforder n + 1 .Then

Proof. Let V ( L n , 1 ) = { u , v 1 , v 2 , ⋯ , v n } and E ( L n , 1 ) = { u i u j | i , j = 1 , 2 , ⋯ , n , i ≠ j } ∪ { v u 1 } . The case for 0 ≤ β ≤ 1 n − 1 , combining Proposition 2.2, we get b β ( L n , 1 ) = 2 . For the case 1 n − 1 < β ≤ 1 , similar as Theorem 3.4, the details omitted here, we have b β ( L n , 1 ) = n .

Next, we determined the IC burning number of the sunflower graph, friendship graph and Dutch windmill graph.

A sunflower S f n is graph with V ( S f n ) = { v , u 1 , u 2 , ⋯ , u n , w 1 , w 2 , ⋯ , w n } and E ( S f n ) = { v u i | i = 1 , 2 , ⋯ , n } ∪ { u i w i | i = 1 , 2 , ⋯ , n }                             ∪ { w i u i + 1 | i = 1 , 2 , ⋯ , n } ∪ { u i u i + 1 | i = 1 , 2 , ⋯ , n } , where edge u n u n + 1 is u n u 1 and w n u n + 1 is w n u 1 , see Figure 2(a).

(a) (b)

Figure 2. S f n with order 2 n + 1 and S f n I with order 3 n + 1 .

An I type sunflower S f n I is a graph with V ( S f n I ) = { v , u 1 , ⋯ , u n , w 1 , ⋯ , w n } and E ( S f n I ) = { v u i | i = 1 , 2 , ⋯ , n } ∪ { v w i | i = 1 , 2 , ⋯ , n } ∪ { v v i | i = 1 , 2 , ⋯ , n }                               ∪ { u i w i | i = 1 , 2 , ⋯ , n } ∪ { u i u i + 1 | i = 1 , 2 , ⋯ , n } , where edge u n u n + 1 is u n u 1 , see Figure 2(b).

A friendship graph F n obtained V ( F n ) = { v , v 1 , v 2 , ⋯ , v n , u 1 , u 2 , ⋯ , u n } and E ( F n ) = { v v i | i = 1 , 2 , ⋯ , n } ∪ { v u i | i = 1 , 2 , ⋯ , n } ∪ { u i v i | i = 1 , 2 , ⋯ , n } , see Figure 3(a).

(a) (b)

Figure 3. F n with order 2 n + 1 and D 4 n with order 3 n + 1 .

A Dutch windmill graph D 4 n satisfied V ( D 4 n ) = { s } ∪ { v i , u i , w i | 1 ≤ i ≤ n } and E ( D 4 n ) = { s u i | i = 1 , 2 , ⋯ , n } ∪ { s v i | i = 1 , 2 , ⋯ , n }                             ∪ { w i u i | i = 1 , 2 , ⋯ , n } ∪ { w i v i | i = 1 , 2 , ⋯ , n } , see Figure 3(b).

Lemma3.6.Foraconnectedgraph G oforder n and δ ( G ) istheminimumdegreeof G .Then, b β ( G ) = n ifandonlyif β > 1 δ ( G ) .

Proof. If β > 1 δ ( G ) , then f ( V ( G ) ) < β . By Proposition 2.1, we have that n ≤ b β ( G ) ≤ n . Therefore, b β ( G ) = n .

If b β ( G ) = n , suppose β ≤ 1 δ ( G ) . Consider the minmum degree of G is δ ( G ) and β ≤ 1 δ ( G ) , then any vertex u of G receives influence from a neighbour is f ( u ) = 1 d ( u ) ≥ β , thus we can selecte n − 1 fire source to burn the whole graph G , a contradiction, thus β > 1 δ ( G ) .

Theorem3.7.Let S f n beasunflowergraphoforder 2 n + 1 , where n ≥ 5 .Then

Proof. Let V ( S f n ) = { v , u 1 , u 2 , ⋯ , u n , w 1 , w 2 , ⋯ , w n } (see Figure 2(a)). The case for 0 ≤ β ≤ 1 5 , by Proposition 2.2, we directly get b β ( S f n ) ≥ 3 . On the other hand, let x 1 = v , x 2 = u 1 and x 3 = w 1 . Clearly, ( x 1 , x 2 , x 3 ) is a β -burning sequence of S f n and thus b β ( S f n ) ≤ 3 . Then, b β ( S f n ) = 3 . Consider the case 1 5 < β ≤ 1 2 , let X = { u i | 1 ≤ i ≤ n } ∪ { v } . Consider f ( X ) < β , by Proposition 2.1, we have that b β ( S f n ) ≥ n + 1 . Now we show that b β ( S f n ) ≤ n + 1 , let x n + 1 = v and x i = u i for 1 ≤ i ≤ n . Clearly, ( x 1 , x 2 , ⋯ , x n + 1 ) is a β -burning sequence of S f n and thus b β ( S f n ) ≤ n + 1 . Thus, b β ( S f n ) = n + 1 . For the case 1 2 < β ≤ 1 , by Lemma 3.6, δ ( G ) = 2 and β > 1 δ ( G ) , thus we directly get b β ( S f n ) = 2 n + 1 .

Theorem3.8.Let S f n I bean I typesunflowergraphoforder 3 n + 1 .Then

Proof. Let V ( S f n I ) = { v , u 1 , u 2 , ⋯ , u n , w 1 , w 2 , ⋯ , w n } . The case for 0 ≤ β ≤ 1 4 , consider d ( v ) = 3 n , by Proposition 2.2, we directly get b β ( S f n I ) = 2 . The case for 1 4 < β ≤ 1 2 , let X = { u i | 1 ≤ i ≤ n } ∪ { v } . Consider f ( X ) < β , by Proposition 2.1, we have that b β ( S f n I ) ≥ n + 1 . Now we show that b β ( S f n ) ≤ n + 1 . let x n + 1 = v and x i = u i for 1 ≤ i ≤ n . Clearly, ( x 1 , x 2 , ⋯ , x n + 1 ) is a β -burning sequence of S f n I and thus b β ( S f n ) ≤ n + 1 . Thus, b β ( S f n I ) = n + 1 . For the case 1 2 < β ≤ 1 , similar as Theorem 3.7. We have b β ( S f n I ) = n + 1 .

Theorem3.9.Let F n beafriendshipgraphoforder 2 n + 1 .Then

Proof. Firstly, this case 0 ≤ β ≤ 1 2 . Consider d ( v ) = 2 n and all f ( V ( F n ) ) ≥ β , by Proposition 2.2, we have that b β ( F n ) = 2 . Now, consider 1 2 < β ≤ 1 . Clearly, if 1 2 < β ≤ 1 , we know that f ( V ( F n ) ) < β . Combine with Lemma 3.6, we directly get b β ( F n ) = 2 n + 1 .

Theorem3.10.Let D 4 n beaDutchwindmillgraphoforder 3 n + 1 .Then

Proof. For the case 0 ≤ β ≤ 1 2 . If n = 1 , by Theorem 3.8, we have that b β ( D 4 n ) = 2 . If n ≥ 2 , let x 1 = v , x 2 = u 1 and x 3 = w 1 . Clearly, ( x 1 , x 2 , x 3 ) is a burning sequence of D 4 n and thus b β ( D 4 n ) ≤ 3 . Consider b β ( D 4 n ) ≥ 3 . When 0 < β ≤ 1 2 , all f ( V ( F n ) ) ≥ β . By Proposition 2.2, we gain b β ( D 4 n ) ≥ 3 and thus b β ( D 4 n ) = 3 . Now, consider the case 1 2 < β ≤ 1 . Clearly, if 1 2 < β ≤ 1 , we know that f ( V ( D 4 n ) ) < β . by Lemma 3.6, we have b β ( D 4 n ) = 3 n + 1 .

Theorem3.11.Foragraph G , let H isaspanningsubgraphof G and ∀ v ∈ H , d H ( v ) = d G ( v ) .Then b β ( G ) ≤ b β ( H ) .

Proof. If β = 0 , then it turns a traditional burning problem, combining Proposition 2.6, we have b ( G ) ≤ b ( H ) . Now we consider 0 < β ≤ 1 , assume b β ( H ) = k , ( x 1 , x 2 , ⋯ , x k ) is a β -burning sequence of H . As f ( v ) = 1 d ( v ) and d H ( v ) = d G ( v ) , therefore, for any vertex v , the influence it receives from a neighbour is f ( v ) = 1 d H ( v ) = 1 d G ( v ) , clearly, ( x 1 , x 2 , ⋯ , x k ) also can be β -burning sequence of G . We get b β ( G ) ≤ k , thus b β ( G ) ≤ b β ( H ) .

This research was supported by NSFC No.12371352.

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