Zepeng Li

On dominating sets of maximal outerplanar and planar graphs

A set D ? V ( G ) of a graph G is a dominating set if every vertex v ? V ( G ) is either in D or adjacent to a vertex in D . The domination number γ ( G ) of a graph G is the minimum cardinality of a dominating set of G . Campos and Wakabayashi (2013) and Tokunaga (2013) proved independently that if G is an n -vertex maximal outerplanar graph having t vertices of degree 2, then γ ( G ) ? n + t 4 . We improve their upper bound by showing γ ( G ) ? n + k 4 , where k is the number of pairs of consecutive 2-degree vertices with distance at least 3 on the outer cycle. Moreover, we prove that γ ( G ) ? 5 n 16 for a Hamiltonian maximal planar graph G with n ? 7 vertices.

Double Roman domination in trees

Abstract A subset S of the vertex set of a graph G is a dominating set if every vertex of G not in S has at least one neighbor in S. The domination number γ ( G ) is defined to be the minimum cardinality among all dominating set of G. A Roman dominating function on a graph G is a function f : V ( G ) → { 0 , 1 , 2 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2 . The weight of a Roman dominating function f is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R ( G ) of G. A double Roman dominating function on a graph G is a function f : V ( G ) → { 0 , 1 , 2 , 3 } satisfying the condition that every vertex u for which f ( u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2 , and every vertex u for which f ( u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2 . The weight of a double Roman dominating function f is the value f ( V ( G ) ) = ∑ u ∈ V ( G ) f ( u ) . The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number γ d R ( G ) of G. Beeler et al. (2016) [6] showed that 2 γ ( G ) ≤ γ d R ( G ) ≤ 3 γ ( G ) and showed that 2 γ ( T ) + 1 ≤ γ d R ( T ) ≤ 3 γ ( T ) for any non-trivial tree T and posed a problem that if it is possible to construct a polynomial algorithm for computing the value of γ d R ( T ) for any tree T. In this paper, we answer this problem by giving a linear time algorithm to compute the value of γ d R ( T ) for any tree T. Moreover, we give characterizations of trees with 2 γ ( T ) + 1 = γ d R ( T ) and γ d R ( T ) + 1 = 2 γ R ( T ) .

On secure domination in trees

Abstract A subset D of the vertex set of a graph G is a secure dominating set of G if D is a dominating set of G and if, for each vertex u not in D, there is a vertex v in D adjacent to u such that the swap set (D \ {v}) ∪ {u} is again a dominating set of G. The secure domination number of G, denoted by γs(G), is the cardinality of a smallest secure dominating set of G. In this paper, we prove that for any tree T on n ≥ 3 vertices, and the bounds are sharp, where ℓ and t are the numbers of leaves and stems of T , respectively. Moreover, we characterize the trees T such that .

A characterization of trees with equal independent domination and secure domination numbers

We give a characterization of trees with equal independent domination and secure domination numbers, which answers a question proposed by Merouane and Chellali (2015).We introduce three operations on trees and prove that any tree with equal independent domination and secure domination number can be obtained by these operations. Let i ( G ) and γ s ( G ) be the independent domination number and secure domination number of a graph G, respectively. Merouane and Chellali (2015) 12 proved that i ( T ) ź γ s ( T ) for any tree T and asked to characterize the trees T with i ( T ) = γ s ( T ) . In this paper, we answer the question. We introduce three operations on trees and prove that any tree T with i ( T ) = γ s ( T ) can be obtained by these operations.

On the signed Roman k-domination: Complexity and thin torus graphs

Abstract A signed Roman k -dominating function on a graph G = ( V ( G ) , E ( G ) ) is a function f : V ( G ) → { − 1 , 1 , 2 } such that (i) every vertex u with f ( u ) = − 1 is adjacent to at least one vertex v with f ( v ) = 2 and (ii) ∑ x ∈ N [ w ] f ( x ) ≥ k holds for any vertex w . The weight of f is ∑ u ∈ V ( G ) f ( u ) , the minimum weight of a signed Roman k -dominating function is the signed Roman k -domination number γ s R k ( G ) of G . It is proved that determining the signed Roman k -domination number of a graph is NP-complete for k ∈ { 1 , 2 } . Using a discharging method, the values γ s R 2 ( C 3 □ C n ) and γ s R 2 ( C 4 □ C n ) are determined for all n .

A note on local coloring of graphs

A local k-coloring of a graph G is a function f : V ( G ) ? { 1 , 2 , ? , k } such that for each S ? V ( G ) , 2 ? | S | ? 3 , there exist u , v ? S with | f ( u ) - f ( v ) | at least the size of the subgraph induced by S. The local chromatic number of G is ? ? ( G ) = min ? { k : G has a local k -coloring } . Chartrand et al. 2 asked: does there exist a graph G k such that ? ? ( G k ) = ? ( G k ) = k ? Furthermore, they conjectured that for every positive integer k, there exists a graph G k with ? ? ( G ) = k such that every local k-coloring of G k uses all of the colors 1 , 2 , ? , k . In this paper we give a affirmative answer to the problem and confirm the conjecture. We study the local k-coloring of graphs.We prove that there exists a graph G k such that ? ? ( G k ) = ? ( G k ) = k for any k. This result gives a affirmative answer to a problem proposed by Chartrand et al. 2.We prove that there exists a graph G k with ? ? ( G ) = k such that every local k-coloring of G k uses all of the colors 1 , 2 , ? , k . This result confirms a conjecture proposed by Chartrand et al. 2.

NP-completeness of local colorings of graphs

We study the local k-coloring of graphs.We prove that it is NP-complete to determine whether a graph has a local k-coloring for fixed k=4 or k=2t1, where t3.We prove that it is NP-complete to determine whether a planar graph has a local 5-coloring even restricted to the maximum degree =7or8. A local k-coloring of a graph G is a function f:V(G){1,2,,k} such that for each SV(G), 2|S|3, there exist u,vS with |f(u)f(v)| at least the size of the subgraph induced by S. The local chromatic number of G is (G)=min{k:Ghas a localk-coloring}. In this paper, we show that it is NP-complete to determine whether a graph has a local k-coloring for fixed k=4 or k=2t1, where t3. In particular, it is NP-complete to determine whether a planar graph has a local 5-coloring even restricted to the maximum degree =7or8.

Acyclic 3-coloring of generalized Petersen graphs

An acyclic

Weak {2}-domination number of Cartesian products of cycles

k