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Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2

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Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2 Linear Separation of Connected Dominating Sets in Graphs (Extended Abstract) Nina Chiarelli1 and Martin Milanicˇ2 1 University of Primorska, UP FAMNIT, Glagoljaskaˇ 8, SI6000 Koper, Slovenia [email protected] 2 University of Primorska, UP IAM, Muzejski trg 2, SI6000 Koper, Slovenia University of Primorska, UP FAMNIT, Glagoljaskaˇ 8, SI6000 Koper, Slovenia [email protected] Abstract are looking for, such as matchings, cliques, stable sets, dominating sets, etc. A connected dominating set in a graph is a dominating set of vertices that induces a The above framework provides a unified way connected subgraph. We introduce and study of describing characteristic properties of several the class of connected-domishold graphs, graph classes studied in the literature, such as which are graphs that admit non-negative real threshold graphs (Chvatal´ and Hammer 1977), weights associated to their vertices such that domishold graphs (Benzaken and Hammer 1978), a set of vertices is a connected dominating total domishold graphs (Chiarelli and Milanicˇ set if and only if the sum of the correspond- 2013a; 2013b) and equistable graphs (Payan 1980; ing weights exceeds a certain threshold. We show that these graphs form a non-hereditary Mahadev, Peled, and Sun 1994). If weights as class of graphs properly containing two above exist and are given with the graph, and the well known classes of chordal graphs: block set T is given by a membership oracle, then a dy- graphs and trivially perfect graphs. We char- namic programming algorithm can be employed acterize, in several ways, the graphs every to find a subset with property P of either maxi- induced subgraph of which is connected- mum or minimum cost (according to a given cost domishold: in terms of forbidden induced function on the vertices/edges) in O(nM) time and subgraphs, in terms of 2-asummability of cer- with M calls of the membership oracle, where n tain derived Boolean functions, and in terms is the number of vertices (or edges) of G and M of the dually Sperner property of certain de- is a given upper bound for (Milanic,ˇ Orlin, and rived hypergraphs. T Rudolf 2011). The framework can also be applied more generally, in the context on Boolean opti- Introduction mization (Milanic,ˇ Orlin, and Rudolf 2011). A possible approach for dealing with the in- In general, the advantages of the above frame- tractability of a given decision or optimization work depend both on the choice of property P and problem is to identify restrictions on input in- on the constraints (if any) imposed on the structure stances under which the problem can still be solved of the set T . For example, if P denotes the property efficiently. One generic framework for describing of being a stable (independent) set, and set T is re- a kind of such restrictions in case of graph prob- stricted to be an interval unbounded from below, lems is the following: Given a graph G, does G we obtain the class of threshold graphs (Chvatal´ admit non-negative integer weights on its vertices and Hammer 1977), which is very well under- (or edges, depending on the problem) and a set T stood and admits several characterizations and lin- of integers such that a subset X of its vertices (or ear time algorithms for recognition and for several edges) has property P if and only if the sum of the optimization problems (see, e.g., (Mahadev and weights of elements of X belongs to T ? Property Peled 1995)). If P denotes the property of being P can denote any of the desired substructures we a dominating set and T is an interval unbounded from above, we obtain the class of domishold mer 2012; Butenko, Kahruman-Anderoglu, and graphs (Benzaken and Hammer 1978), which en- Ursulenko 2011; Chandran et al. 2012; Duckworth joy similar properties as the threshold graphs. On and Mans 2009; Fomin, Grandoni, and Kratsch the other hand, if P is the property of being a 2008; Karami et al. 2012; Schaudt 2012a; 2012b; maximal stable set and T is restricted to consist Schaudt and Schrader 2012). of a single number, we obtain the class of equi- Definition 1. A graph G = (V; E) is said to stable graphs (Payan 1980), for which the recogni- be connected-domishold (c-domishold for short) tion complexity is open (see, e.g., (Levit, Milanic,ˇ if there exists a pair (w; t) where w : V R+ and Tankus 2012)), no structural characterization ! is a weight function and t R+ is a thresh- is known, and several NP-hard optimization prob- old such that for every subset2S V , w(S) := lems remain intractable on this class (Milanic,ˇ Or- P w(x) t if and only if⊆ S is a con- lin, and Rudolf 2011). x2S nected dominating≥ set in G. A pair (w; t) as above Notions and results from the theory of Boolean will be referred to as a connected-domishold (c- functions (Crama and Hammer 2011) and hyper- domishold) structure of G. graph theory (Berge 1989) can be useful for the Observe that if G is disconnected, then G does study of graph classes defined within the above not have any c-dominating sets and is thus trivially framework. For instance, the characterization of c-domishold (just set w(x) = 1 for all x V (G) hereditary total domishold graphs in terms of for- and t = V (G) + 1). 2 bidden induced subgraphs from (Chiarelli and Mi- j j lanicˇ 2013b) is based on the facts that every Example 1. The complete graph of order n is threshold Boolean function is 2-asummable (Chow c-domishold. Indeed, any nonempty subset S ⊆ 1961) and that every dually Sperner hypergraph V (Kn) is a connected dominating set of Kn, and is threshold (Chiarelli and Milanicˇ 2013a). More- the pair (w; 1) where w(x) = 1 for all x V (Kn) 2 over, the fact that threshold Boolean functions are is a c-domishold structure of Kn. closed under dualization and can be recognized Example 2. The 4-cycle C4 is not c-domishold: in polynomial time (Peled and Simeone 1985) Denoting its vertices by v1; v2; v3; v4 in the cyclic leads to efficient algorithms for recognizing to- order, we see that a subset S V (C4) is c- tal domishold graphs, and for finding a minimum domishold if and only if it contains⊆ an edge. There- total dominating set in a given total domishold fore, if (w; t) is a c-domishold structure of C4, graph (Chiarelli and Milanicˇ 2013a). then w(v ) + w(v ) t for all i 1; 2; 3; 4 i i+1 ≥ 2 f g It is the aim of this note to present another (indices modulo 4), which implies w(V (C4)) ≥ application of the notions of threshold Boolean 2t. On the other hand, w(v1) + w(v3) < t and functions/hypergraphs to the above graph theo- w(v2) + w(v4) < t, implying w(V (C4)) < 2t. retic framework. More specifically, we introduce Example 3. The graph G obtained from C4 by and study the case when P is the property of be- adding to it a new vertex, say v5, and making it ing a connected dominating set and T is an inter- adjacent exactly to one vertex of the C4, say to val unbounded from above. Given a graph G = v4, is c-domishold: the (inclusion-wise) minimal c- (V; E), a connected dominating set (c-dominating dominating sets of G are v1; v4 and v3; v4 , set for short) is a subset S of the vertices of G hence a c-domishold structuref ofgG isf given byg that is dominating, that is, every vertex of G is ei- w(v2) = w(v5) = 0, w(v1) = w(v3) = 1, ther in S or has a neighbor in S, and connected, w(v4) = 2, and t = 3. that is, the subgraph of G induced by S, hence- forth denoted by G[S], is connected. The notion A graph class is said to be hereditary if it is of connected dominating sets in graphs is one of closed under vertex deletion. The above exam- the many variants of domination. It finds applica- ples show that, contrary to the classes of thresh- tions in modeling wireless network connectivity, old and domishold graphs, the class of connected- and has been extensively studied in the literature, domishold graphs is not hereditary. This motivates see, e.g., the books (Du and Wan 2013; Haynes, the following definition: Hedetniemi, and Slater 1998b; 1998a), and re- Definition 2. A graph G is said to be hereditary cent papers (Ananchuen, Ananchuen, and Plum- connected-domishold (hereditary c-domishold for short) if every induced subgraph of it is connected- A transversal of is a set S such that domishold. S e = for all eH . A hypergraph⊆ V = ( ; ) \ 6 ; 2 E H V E In general, for a graph property Π, we will say is threshold if there exist a weight function w : R+ and a threshold t R+ such that for all that a graph is hereditary Π if every induced sub- V! 2 graph of it satisfies Π. subsets X , it holds that w(X) t if and only if X contains⊆ V no edge of (Golumbic≤ 2004). A Our main result (Theorem 2) is a characteriza- H tion of hereditary connected-domishold graphs in hypergraph = ( ; ) is said to be Sperner (or: a clutter) if noH edgeV ofE contains another edge, or, terms of forbidden induced subgraphs, in terms of H 2-asummability of certain derived Boolean func- equivalently, if for every two distinct edges e and f of , it holds that min e f ; f e 1 : tions, and in terms of the dually Sperner property H fj n j j n jg ≥ of certain derived hypergraphs.
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