Nordic Journal of Computing 1(1994), 157{171. COMPLEXITY OF DOMINATION-TYPE PROBLEMS IN GRAPHS∗ JAN ARNE TELLE Department of Computer and Information Science University of Oregon, Eugene, Oregon 97403, USA [email protected] Abstract. Many graph parameters are the optimal value of an objective function over selected subsets S of vertices with some constraint on how many selected neigh- bors vertices in S, and vertices not in S, can have. Classic examples are minimum dominating set and maximum independent set. We give a characterization of these graph parameters that unifies their definitions, facilitates their common algorithmic treatment and allows for their uniform complexity classification. This characteriza- tion provides the basis for a taxonomy of domination-type and independence-type problems. We investigate the computational complexity of problems within this taxonomy, identify classes of NP -complete problems and classes of problems solv- able in polynomial time. CR Classification: F.2.2, G.2.2 1. Introduction If every vertex in a selected subset S of vertices of a graph has zero selected neighbors then S is an independent set, and similarly if every vertex not in S has at least one selected neighbor then S is a dominating set. This suggests a common characterization of independent sets and dominating sets based on the constraints imposed on the number of selected neighbors the vertices in S, and vertices not in S, can have. As we show in this paper, a large collection of well-known vertex subset properties admit such a characteriza- tion. For these vertex subset properties we consider the graph parameters definable by optimization over, or existence of, vertex subsets having the property. From the standard definitions of these parameters it is not ob- vious that they are related as described here. This characterization thus facilitates the common algorithmic treatment of the problems computing these parameters. In recent years, a variety of domination-type parameters in graphs have been introduced, and the number of papers devoted to this topic is steadily increasing [11, 12]. ∗ This work supported by The Norwegian Research Council for Science and Humanities and by NSF grant CCR9213439. Received July 1993. Accepted December 1993. 158 JAN ARNE TELLE In the next section, we present our characterization and show that many of the graph parameters found in the literature admit such a characteri- zation. We give a table cataloging the computational complexity of com- puting these parameters. In section 3, we investigate the computational complexity of further problems admitting the characterization. We identify several classes of NP -complete and of polynomial-time solvable problems. The NP -completeness results identify properties with interesting features, such as when both maximum and minimum versions are NP -complete (in- dependent dominating sets), or when merely deciding whether a graph has a vertex subset with the property is NP -complete (perfect codes). A natural by-product of these results is the introduction of several new domination- type parameters in graphs. 2. Characterization of Domination-type Problems We use standard graph terminology [3]. For a vertex v 2 V (G) of a graph G, let NG(v) = fu : (u; v) 2 E(G)g be the set of neighbors of v and degG(v) = jNG(v)j. For S ⊆ V (G) let G[S] denote the graph induced in G by S and let the symbols σ and ρ indicate membership in S and membership in V (G) n S = fv 2 V (G) : v 62 Sg, respectively. Definition 1. Given a graph G and a set S ⊆ V (G) of selected vertices ◦ The state of a vertex v 2 V (G) is df ρi if v 62 S and jNG(v) \ Sj = i stateS(v) = σi if v 2 S and jNG(v) \ Sj = i ◦ Define syntactic abbreviations ρ≤i ≡ ρ0; ρ1; :::; ρi ρ≥i ≡ ρi; ρi+1; ::: σ≤i ≡ σ0; σ1; :::; σi σ≥i ≡ σi; σi+1; ::: Thus ρ≥i and σ≥i each represents an infinite set of states. Mnemonically, σ represents a vertex selected for S and ρ a vertex rejected from S, with the subscript indicating the number of neighbors the vertex has in S. A variety of vertex subset properties can be defined by allowing only a specific set L as legal states of vertices. For example, S is a dominating set if state ρ0 is not allowed for any vertex, giving the legal states L = fρ≥1; σ≥0g. Optimization problems over these sets often maximize or minimize the cardinality of the set of vertices with states in a given M ⊆ L. For instance, in the minimum dominating set problem, M = fσ≥0g. COMPLEXITY OF DOMINATION-TYPE PROBLEMS IN GRAPHS 159 Our notation Standard terminology 9[L] max[L] min[L] [ρ≥0; σ0]-set Independent set P NPC P [ρ≥1; σ≥0]-set Dominating set P P NPC [ρ≤1; σ0]-set Strong Stable set or 2-Packing P NPC P [ρ1; σ0]-set Efficient Dominating set or Perfect Code NPC NPC NPC [ρ≥1; σ0]-set Independent Dominating set P NPC NPC [ρ1; σ≥0]-set Perfect Dominating set P P NPC [ρ≥1; σ≥1]-set Total Dominating set P P NPC [ρ1; σ1]-set Total Perfect Dominating set NPC NPC NPC [ρ≤1; σ≥0]-set Nearly Perfect set P P P [ρ≤1; σ≤1]-set Total Nearly Perfect set P NPC P [ρ1; σ≤1]-set Weakly Perfect Dominating set NPC NPC NPC [ρ≥0; σ≤q]-set Induced Bounded-Degree subgraph (q ≥ 0) P NPC P [ρ≥q; σ≥0]-set q-Dominating set (q ≥ 1) P P NPC [ρ≥0; σq ]-set Induced q-Regular subgraph (q ≥ 0) P NPC P Table I: Some vertex subset properties and the complexity of derived problems. Our notation Standard terminology Complexity 9[ρ1; σ0] Perfect Code Problem NPC min[ρ≥1; σ≥0] Minimum Dominating Set Problem NPC max[ρ≥0; σ0] Maximum Independent Set Problem NPC minfρ≥0g[ρ≥0; σ0] Minimum Vertex Cover Problem NPC maxfρ1g[ρ≥0; σ≥0] Efficiency Problem NPC Table II: Examples of graph problems. Definition 2. Given sets M and L of vertex states and a graph G: ◦ S ⊆ V (G) is an [L]-set if 8v 2 V (G) : stateS(v) 2 L ◦ 9[L] is the problem deciding whether there exists any [L]-set S ⊆ V (G) ◦ minM[L] (or maxM[L]) is the problem minimizing (or maximizing) jfv : stateS(v) 2 Mgj over all [L]-sets S ⊆ V (G) ◦ min[L] (or max[L]) is shorthand for minM[L] (or maxM[L]) when M consists of all σ-states in L, in effect optimizing the cardinality of the selected set of vertices. Thus, a dominating set is a [ρ≥1; σ≥0]-set, with the square brackets im- plying the set notation. Table I shows some of the classical vertex subset properties and also the complexity of derived problems, with P denoting Polynomial time and NPC denoting NP -Complete. Most of these complex- ity results are old [10, 5, 8, 1, 6, 9, 4, 13], and others are among the results given in the next section. Properties traditionally defined using closed neigh- borhoods are easily captured by the characterization. Table II shows exam- ples of graph problems [2, 18] expressed using the characterization. Note that complementary problems, e.g. Maximum Independent Set and Mini- mum Vertex Cover, are both expressible. Table I can be used as a quick reference guide to the exact definitions of the various properties represented 160 JAN ARNE TELLE and to the complexity of the associated problems. The characterization may also be useful when introducing new parameters. 3. Complexity Results We are mainly interested in classifying problems admitting the given charac- terization as NP -complete or as solvable in polynomial time. The objective functions most studied in the past involve minimizing or maximizing the cardinality of the set of selected vertices, and for each entry in Table I, ex- cept Nearly Perfect Sets, there is at least one NP -complete problem related to such a parameter. In this paper we continue this trend, and the optimiza- tion problems we concentrate on are of the form min[L] and max[L]. For certain subset properties, such as Perfect Code, it is well known that even deciding whether a graph has any such set is an NP -complete problem. In the following Lemma we observe several consequences of NP -completeness of an 9[L] problem. Lemma 1. If 9[L] is NP -complete on a class of graphs C then any decision problems of the form max[L], min[L], maxM[L], minM[L] or maxL[Q], L ⊆ Q are NP -complete on C. Conversely, if any of the latter problems have a polynomial time algorithm, then so does 9[L]. Proof. The decision version of maxM[L] takes a graph G and an integer k as input, and asks if G has an [L]-set S with jfv : stateS(v) 2 Mgj ≥ k. Thus, with an algorithm for the decision version of maxM[L], we can decide 9[L] by a single call of that algorithm providing the integer k = 0 as the second part of the input. With an algorithm for minM[L] or maxL[Q] problems we decide 9[L] using k = jV (G)j for the input graph G as the second part of the input. 2 In particular, Theorems 1,2,3,4 and 7 can each be combined with Lemma 1 to yield corollaries of this kind. We will not state these corollaries explic- itly. We observe from Table I that vertex subset properties attracting most interest in the past are characterizable by two syntactic states (using the abbreviations) in which vertices have zero, one, at least zero, or at least one selected neighbors.