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Let G be a graph with vertex set V. A subset S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number is the minimum cardinality of a semipaired dominating set of G. We characterize the trees having a unique minimum semipaired dominating set. We also determine an upper bound on the semipaired domination number of these trees and characterize the trees attaining this bound.

In this paper we present a brief survey of bounds on selected domination parameters. We focus primarily on bounds on domination parameters in terms of the order and minimum degree of the graph. We present a list of open problems and conjectures that have yet to be solved in the hope of attracting future researchers to the field.

A set S of vertices in a graph G is a dominating set of G if every vertex not in S is adjacent to some vertex in S. The domination number, γ(G), of G is the minimum cardinality of a dominating set of G. The authors proved in [A new lower bound on the domination number of a graph, J. Comb. Optim. 38 (2019) 721–738] that if G is a connected graph of order n ≥ 2 with k ≥ 0 cycles and ℓ leaves, then γ(G) ≥ ⌈(n − ℓ + 2 − 2k)/3⌉. As a consequence of the above bound, γ(G) = (n − ℓ + 2(1 − k) + m)/3 for some integer m ≥ 0. In this paper, we characterize the class of cactus graphs achieving equality here, thereby providing a classification of all cactus graphs according to their domination number.

A set D of vertices in a graph G is a dominating set of G if every vertex outside D is adjacent in G to some vertex in D. A set D of vertices in G is a semitotal dominating set of G if D is a dominating set of G and every vertex in D is within distance 2 from another vertex of D. Given a graph G and a positive integer k, the semitotal domination problem is to decide whether G has a semitotal dominating set of cardinality at most k. The semitotal domination problem is known to be NP-complete for chordal graphs and bipartite graphs as shown in [M.A. Henning and A. Pandey, Algorithmic aspects of semitotal domination in graphs, Theoret. Comput. Sci. 766 (2019) 46–57]. In this paper, we present a linear time algorithm to compute a minimum semitotal dominating set in block graphs. On the other hand, we show that the semitotal domination problem remains NP-complete for undirected path graphs.

A subset D ⊆ VG is a dominating set of G if every vertex in VG – D has a neighbor in D, while D is a paired-dominating set of G if D is a dominating set and the subgraph induced by D contains a perfect matching. A graph G is a DPDP -graph if it has a pair (D, P) of disjoint sets of vertices of G such that D is a dominating set and P is a paired-dominating set of G. The study of the DPDP -graphs was initiated by Southey and Henning [Cent. Eur. J. Math. 8 (2010) 459–467; J. Comb. Optim. 22 (2011) 217–234]. In this paper, we provide conditions which ensure that a graph is a DPDP -graph. In particular, we characterize the minimal DPDP -graphs.

We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing’s conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: γ (X□Y) ≥ max γ(X□Y)≥max{12γ(X)γt(Y),12γt(X)γ(Y)}\gamma \left( {X \square Y} \right) \ge \max \left\{ {{1 \over 2}\gamma (X){\gamma _t}(Y),{1 \over 2}{\gamma _t}(X)\gamma (Y)} \right\}, where γ stands for the domination number, γt is the total domination number, and X□Y is the Cartesian product of graphs X and Y.

A self-complementary graph is a graph isomorphic to its complement. A set S of vertices in a graph G is a restrained dominating set if every vertex in V(G) \ S is adjacent to a vertex in S and to a vertex in V(G) \ S. The restrained domination number of a graph G is the minimum cardinality of a restrained dominating set of G. In this paper, we study restrained domination in self-complementary graphs. In particular, we characterize the self-complementary graphs having equal domination and restrained domination numbers.

For an integer k at least 2, and a graph G, let fk(G) be the minimum cardinality of a set X of vertices of G such that G − X has either k vertices of maximum degree or order less than k. Caro and Yuster [Discrete Mathematics 310 (2010) 742–747] conjectured that, for every k, there is a constant ck such that fk(G)≤ckn(G){f_k}\left( G \right) \le {c_k}\sqrt {n\left( G \right)} for every graph G. Verifying a conjecture of Caro, Lauri, and Zarb [arXiv:1704.08472v1], we show the best possible result that, if t is a positive integer, and F is a forest of order at most 16(t3+6t2+17t+12){1 \over 6}\left( {{t^3} + 6{t^2} + 17t + 12} \right), then f2(F ) ≤ t. We study f3(F ) for forests F in more detail obtaining similar almost tight results, and we establish upper bounds on fk(G) for graphs G of girth at least 5. For graphs G of girth more than 2p, for p at least 3, our results imply fk(G)=O(n(G)p+13p){f_k}\left( G \right) = O\left( {n{{\left( G \right)}^{{{p + 1} \over {3p}}}}} \right) . Finally, we show that, for every fixed k, and every given forest F , the value of fk(F ) can be determined in polynomial time.

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The minimum cardinality of a total forcing set in G is its total forcing number, denoted Ft(G). We prove that if T is a tree of order n ≥ 3 with maximum degree Δ and with n1 leaves, then n1≤Ft(T)≤1Δ((Δ-1)N+1){n_1} \le {F_t}\left( T \right) \le {1 \over \Delta }\left( {\left( {\Delta - 1} \right)N + 1} \right). In both lower and upper bounds, we characterize the infinite family of trees achieving equality. Further we show that Ft(T ) ≥ F (T ) + 1, and we characterize the extremal trees for which equality holds.

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number, i(G), of G is the minimum cardinality of an independent dominating set. Goddard and Henning [Discrete Math. 313 (2013) 839–854] posed the conjecture that if G ∉ {K3,3, C5 □ K2} is a connected, cubic graph on n vertices, then i(G)≤38n$i\left( G \right) \le {3 \over 8}n$, where C5 □ K2 is the 5-prism. As an application of known result, we observe that this conjecture is true when G is 2-connected and planar, and we provide an infinite family of such graphs that achieve the bound. We conjecture that if G is a bipartite, planar, cubic graph of order n, then i(G)≤13n$i\left( G \right) \le {1 \over 3}n$, and we provide an infinite family of such graphs that achieve this bound.

A dominating set of a graph G is a subset D ⊆ VG such that every vertex not in D is adjacent to at least one vertex in D. The cardinality of a smallest dominating set of G, denoted by γ(G), is the domination number of G. The accurate domination number of G, denoted by γa(G), is the cardinality of a smallest set D that is a dominating set of G and no |D|-element subset of VG \ D is a dominating set of G. We study graphs for which the accurate domination number is equal to the domination number. In particular, all trees G for which γa(G) = γ(G) are characterized. Furthermore, we compare the accurate domination number with the domination number of different coronas of a graph.

Let G be a graph with vertex set V and no isolated vertices. A sub-set S ⊆ V is a semipaired dominating set of G if every vertex in V \ S is adjacent to a vertex in S and S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number γpr2(G) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a connected graph G of order n ≥ 3, then γpr2(G)≤23n${\gamma _{{\rm{pr}}2}}(G) \le {2 \over 3}n$, and we characterize the extremal graphs achieving equality in the bound.

A subset S of vertices of a graph G is a dominating set of G if every vertex not in S has a neighbor in S, while S is a total dominating set of G if every vertex has a neighbor in S. If S is a dominating set with the additional property that the subgraph induced by S contains a perfect matching, then S is a paired-dominating set. The domination number, denoted γ(G), is the minimum cardinality of a dominating set of G, while the minimum cardinalities of a total dominating set and paired-dominating set are the total domination number, γt(G), and the paired-domination number, γpr(G), respectively. For k ≥ 2, let G be a connected k-regular graph. It is known [Schaudt, Total domination versus paired domination, Discuss. Math. Graph Theory 32 (2012) 435–447] that γpr(G)/γt(G) ≤ (2k)/(k+1). In the special case when k = 2, we observe that γpr(G)/γt(G) ≤ 4/3, with equality if and only if G ≅ C5. When k = 3, we show that γpr(G)/γt(G) ≤ 3/2, with equality if and only if G is the Petersen graph. More generally for k ≥ 2, if G has girth at least 5 and satisfies γpr(G)/γt(G) = (2k)/(k + 1), then we show that G is a diameter-2 Moore graph. As a consequence of this result, we prove that for k ≥ 2 and k ≠ 57, if G has girth at least 5, then γpr(G)/γt(G) ≤ (2k)/(k +1), with equality if and only if k = 2 and G ≅ C5 or k = 3 and G is the Petersen graph.

A dominating set in a graph G is a set S of vertices such that every vertex in V (G) \ S is adjacent to at least one vertex in S, and the domination number of G is the minimum cardinality of a dominating set of G. Placing constraints on a dominating set yields different domination parameters, including total, connected, restrained, and clique domination numbers. In this paper, we study relationships among domination parameters of a graph and its complement.

We characterize a large subclass of the class of those graphs G for which the exponential domination number of H equals the domination number of H for every induced subgraph H of G.

A graph G is a DTDP-graph if it has a pair (D, T) of disjoint sets of vertices of G such that D is a dominating set and T is a total dominating set of G. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-graphs without loops. [ABSTRACT FROM AUTHOR]

In the domination game on a graph G, the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices dominated. This process eventually produces a dominating set of G; Dominator aims to minimize the size of this set, while Staller aims to maximize it. The size of the dominating set produced under optimal play is the game domination number of G, denoted by γg(G). Kinnersley, West and Zamani [SIAM J. Discrete Math. 27 (2013) 2090-2107] posted their 3/5-Conjecture that γg(G) ≤ ⅗n for every isolate-free forest on n vertices. Brešar, Klavžar, Košmrlj and Rall [Discrete Appl. Math. 161 (2013) 1308-1316] presented a construction that yields an infinite family of trees that attain the conjectured 3/5-bound. In this paper, we provide a much larger, but simpler, construction of extremal trees. We conjecture that if G is an isolate-free forest on n vertices satisfying γg(G) = ⅗n, then every component of G belongs to our construction.

A set D of vertices in an isolate-free graph is a total dominating set if every vertex is adjacent to a vertex in D. If the set D has the additional property that the subgraph induced by D contains a perfect matching, then D is a paired dominating set of G. The total domination number γt(G) and the paired domination number γpr(G) of a graph G are the minimum cardinalities of a total dominating set and a paired dominating set of G, respectively. The total domination stability (respectively, paired domination stability) of G, denoted stγt(G) (respectively, stγpr(G)), is the minimum size of a non-isolating set of vertices in G whose removal changes the total domination number (respectively, paired domination number). In this paper, we study total and paired domination stability in prisms. [ABSTRACT FROM AUTHOR]

A set S of vertices of a graph G is a dominating set if every vertex not in S is adjacent to a vertex of S and is a total dominating set if every vertex of G is adjacent to a vertex of S. The cardinality of a minimum dominating (total dominating) set of G is called the domination (total domination) number. A set that does not dominate (totally dominate) G is called a non-dominating (non-total dominating) set of G. A partition of the vertices of G into non-dominating (non-total dominating) sets is a non-dominating (non-total dominating) set partition. We show that the minimum number of sets in a non-dominating set partition of a graph G equals the total domination number of its complement G̅ and the minimum number of sets in a non-total dominating set partition of G equals the domination number of G̅ . This perspective yields new upper bounds on the domination and total domination numbers. We motivate the study of these concepts with a social network application.

Let H = (V, E) be a hypergraph with vertex set V and edge set E. A dominating set in H is a subset of vertices D ⊆ V such that for every vertex v ∈ V \ D there exists an edge e ∈ E for which v ∈ e and e ∩ D ≠ ∅. The domination number γ(H) is the minimum cardinality of a dominating set in H. It is known [Cs. Bujtás, M.A. Henning and Zs. Tuza, Transversals and domination in uniform hypergraphs, European J. Combin. 33 (2012) 62-71] that for k ≥ 5, if H is a hypergraph of order n and size m with all edges of size at least k and with no isolated vertex, then γ(H) ≤ (n + ⌊(k − 3)/2⌋m)/(⌊3(k − 1)/2⌋). In this paper, we apply a recent result of the authors on hypergraphs with large transversal number [M.A. Henning and C. Löwenstein, A characterization of hypergraphs that achieve equality in the Chvátal-McDiarmid Theorem, Discrete Math. 323 (2014) 69-75] to characterize the hypergraphs achieving equality in this bound.

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating set of G. We observe that γ(G) ≤ γt2(G) ≤ γt(G). We characterize the set of vertices that are contained in all, or in no minimum semitotal dominating set of a tree.

Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, γtd(G)$\gamma _t^d (G)$ , is the minimum cardinality of such a set. We observe that γtd(G)≤γt(G)$\gamma _t^d (G) \le \gamma _t (G)$ . A leaf of G is a vertex of degree 1, while a support vertex of G is a vertex adjacent to a leaf. We show that if T is a tree of order n with ℓ leaves and s support vertices, then 2(n−ℓ+3)/5≤γtd(T)≤(n+s−1)/2$2(n - \ell + 3)/5 \le \gamma _t^d (T) \le (n + s - 1)/2$ and we characterize the families of trees which attain these bounds. For every tree T, we show have γt(T)/γtd(T)

In a graph G, a vertex is said to 2-step dominate itself and all the vertices which are at distance 2 from it in G. A set D of vertices in G is called a hop dominating set of G if every vertex outside D is 2-step dominated by some vertex of D. Given a graph G and a positive integer k, the hop domination problem is to decide whether G has a hop dominating set of cardinality at most k. The hop domination problem is known to be NP-complete for bipartite graphs. In this paper, we design a linear time algorithm for computing a minimum hop dominating set in chordal bipartite graphs. [ABSTRACT FROM AUTHOR]

If S is a set of colored vertices in a simple graph G, then one may allow a colored vertex with exactly one non-colored neighbor to force its noncolored neighbor to become colored. If by iteratively applying this color change rule, all of the vertices in G become colored, then S is a zero forcing set of G. The minimum cardinality of a zero forcing set in G, written Z(G), is the zero forcing number of G. If in addition, S induces a subgraph of G without isolated vertices, then S is a total forcing set of G. The total forcing number of G, written Ft(G), is the minimum cardinality of a total forcing set in G. In this paper we introduce, and study, the notion of graphs for which all vertices are contained in some minimum zero forcing set, or some minimum total forcing set; we call such graphs ZF-dense and TF-dense, respectively. A graph is ZTF-dense if it is both ZF-dense and TF-dense. We determine various classes of ZTF-dense graphs, including among others, cycles, complete multipartite graphs of order at least three that are not stars, wheels, n-dimensional hypercubes, and diamond-necklaces. We show that no tree of order at least three is ZTF-dense. We show that if G and H are connected graphs of order at least two that are both ZF-dense, then the join G + H of G and H is ZF-dense. [ABSTRACT FROM AUTHOR]

A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the graph.