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 subset∈S 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). x∈S 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). ∈ bidden induced subgraphs from (Chiarelli and Mi- | | 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) ∈ 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 ≥ ∈ { } 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 structure{ of}G is{ given by} 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 ∅ ∈ 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 → ∈ 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 {| \ | | \ |} ≥ of certain derived hypergraphs. Before stating the In (Chiarelli and Milanicˇ 2013a), the following result, we give in the next section some prelimi- class of threshold hypergraphs was introduced. nary definitions and results. Definition 3. (Chiarelli and Milanicˇ 2013a) A hy- pergraph = ( , ) is said to be dually Sperner Preliminary results if for everyH two distinctV E edges e and f of , it holds H Boolean functions. Let n be positive integer. that min e f , f e 1 . {| \ | | \ |} ≤ Given two vectors x, y 0, 1 n, we write x y ∈ { } ≤ Lemma 1. (Chiarelli and Milanicˇ 2013a) Every if xi yi for all i [n] = 1, . . . , n . A Boolean dually Sperner hypergraph is threshold. function≤ f : 0, 1∈ n { 0, 1 is}positive (or: { } → { } monotone) if f(x) f(y) holds for every two The main notion that will provide the link be- ≤n vectors x, y 0, 1 such that x y. A positive tween threshold Boolean functions and hyper- ∈ { } ≤ Boolean function f is said to be threshold if there graphs is that of separators in graphs. A separator exist non-negative real weights w = (w1, . . . , wn) in a graph G = (V,E) is a subset S V (G) such and a non-negative real number t such that for that G S is not connected. A separator⊆ is minimal every x 0, 1 n, f(x) = 0 if and only if if it does− not contain any other separator. The fol- Pn ∈ { } wixi t. Such a pair (w, t) is called a lowing characterization of c-dominating sets was i=1 ≤ separating structure of f. Every threshold posi- proved in (Kante´ et al. 2012). tive Boolean function admits an integral separating structure. Proposition 1. (Kante´ et al. 2012) In every con- Threshold Boolean functions have been charac- nected graph G = (V,E), a subset D V is a c-dominating set if and only if D S = ⊆for every terized in (Chow 1961) and (Elgot 1960), as fol- ∩ 6 ∅ lows. For k 2, a positive Boolean function f : minimal separator S in G. 0, 1 n ≥0, 1 is said to be k-summable if, for In other words, c-dominating sets are exactly some{ }r → {2, . .} . , k , there exist r (not necessar- the transversals of the minimal separators. Based ily distinct)∈ { false points} of f, say, x1, x2, . . . , xr, on this fact and the fact that threshold Boolean and r (not necessarily distinct) true points of f, functions are closed under dualization (Crama and 1 2 r Pr i Pr i say y , y , . . . , y , such that i=1 x = i=1 y . Hammer 2011), a similar technique as that used (A false point of f is an input vector x 0, 1 n to prove Proposition 4.1 in (Chiarelli and Milanicˇ such that f(x) = 0; a true point is defined∈ { anal-} 2013b) can be used to characterize c-domishold ogously.) Function f is said to be k-asummable if graphs in terms of the thresholdness properties of it is not k-summable, and it is asummable if it is a derived Boolean function and of a derived hy- k-asummable for all k 2. pergraph. Given a graph G = (V,E), its minimal ≥ Theorem 1 ((Chow 1961), (Elgot 1960), see also separator function is the positive Boolean func- V Theorem 9.14 in (Crama and Hammer 2011)). A tion msG : 0, 1 0, 1 that takes value 1 { } → { } V positive Boolean function is threshold if and only precisely on vectors x 0, 1 whose support f ∈ { } if it is asummable. set S(x) := v V : xv = 1 contains some minimal separator{ ∈ of G. The hypergraph} of mini- Hypergraphs. A hypergraph is a pair = ( , ) mal separators of G is the hypergraph HmS(G) = where is a finite set of vertices andH isV a setE (V (G), (G)), where (G) = S : S V (G) of subsetsV of , called (hyper)edges (BergeE 1989). and S isS a minimal separatorS in G{ . ⊆ V } Proposition 2. For a connected graph G = Lemma 2. (Kumar and Madhavan 1998) If S is a (V,E), the following are equivalent: minimal separator of a chordal graph G, then each connected component of has a vertex that is 1. G is c-domishold. G S adjacent to all the vertices of\ S. 2. Its minimal separator function msG of G is threshold. Characterizations of hereditary 3. Its hypergraph of minimal separators HmS(G) is threshold. c-domishold graphs Now we state our main result. Due to space con- Moreover, if (w1, . . . , wn, t) is an integral sep- straints, we only give a part of the proof here. arating structure of msG or of HmS(G), then Pn Theorem 2. For every graph G, the following are (w; i=1 wi t) with w(vi) = wi for all i [n] is a c-domishold− structure of G. ∈ equivalent: 1. G is hereditary c-domishold. Definition 4. A graph G is ms-dually-Sperner if 2. G is hereditary ms-2-asummable. its hypergraph of minimal separators HmS(G) is 3. G is hereditary ms-dually-Sperner. dually Sperner, and ms-2-asummable if its minimal 4. G is a F1,F2,H1,H2,... -free chordal graph, separator function msG is 2-asummable. { } where the graphs F1, F2, and a general member Proposition 2, Theorem 1 and Lemma 1 imply of the family Hi are depicted in Fig. 1. the following result. { } Proposition 3. Every ms-dually-Sperner graph is 1 2 3 i c-domishold. Every c-domishold graph is ms-2- F F Hi (i 1) asummable. 1 2 ≥ Neither of the two statements in Proposition 3 Figure 1: Graphs F , F , and H . can be reversed. The reader can easily verify that 1 2 i the graph obtained from the complete graph K 4 We now examine some of the consequences of by gluing a triangle on every edge is c-domishold the forbidden induced subgraph characterization graph but not ms-dually-Sperner. Moreover, there of hereditary c-domishold graphs given by Theo- exists an ms-2-asummable graph G that is not rem 2. The diamond and the kite (also known as c-domishold. This can be derived using the fact the co-fork or the co-chair) are the graphs depicted that not every 2-asummable positive Boolean func- in Fig. 2. tion is threshold (Theorem 9.15 in (Crama and Hammer 2011)), results from (Chiarelli and Mi- lanicˇ 2013b) establishing the connections between threshold Boolean functions and total domishold split graphs, and the observation that a split graph diamond kite without universal vertices is c-domishold if and only if it is total domishold. (A graph is split if its Figure 2: The diamond and the kite. vertex set can be partitioned into a clique and an independent set, where a clique is a set of pairwise The equivalence between items (1) and (4) in adjacent vertices, and an independent set is a set of Theorem 2 implies that the class of hereditary c- pairwise non-adjacent vertices.) domishold graphs is a proper generalization of the Our main result (Theorem 2 below) implies a class of kite-free chordal graphs. partial converse of Proposition 3: both statements can be reversed if we require the properties to Corollary 1. Every kite-free chordal graph is hold in the stronger, hereditary sense. Our proof hereditary c-domishold. of Theorem 2 will rely on the following property Furthermore, Corollary 1 implies that the class of chordal graphs. Recall that a graph G is chordal of hereditary c-domishold graphs generalizes two if it does not contain any induced cycle of order at well known classes of chordal graphs, the block least 4. graphs and the trivially perfect graphs. A graph is said to be a block graph if every block of it dually-Sperner, there exist two minimal separators is complete. The block graphs are well known to in G, say S and S0, such that min S , S0 2. coincide with the diamond-free chordal graphs. A Let C = a, b for some a, b S {|S|0 with| |}a ≥= b graph G is said to be trivially perfect (Golumbic and let C0{= }a0, b0 for some∈a0, b\0 S0 S with6 1978) if for every induced subgraph H of G, it a0 = b0. Further,{ let} X, Y be two components∈ \ of holds α(H) = (H) , where α(H) denotes the G 6 S and X0, Y 0 two components of G S0. By independence number|C of| H (that is, the maximum Lemma− 2, there exist vertices x X and−y Y size of an independent set in H), and (H) de- such that each of x and y dominates∈ S and x0 ∈X0 notes the set of all maximal cliques of HC. Trivially and y0 Y 0 such that each of x0 and y0 dominates∈ perfect graphs coincide with the so-called quasi- S0. Define∈ Z = x, y and Z0 = x0, y0 . threshold graphs (Yan, Chen, and Chang 1996), { } { } 0 0 and also with the P4,C4 -free graphs (Golumbic Claim 1. Either N(x) a , b = or N(y) 0 0 ∩ { } 0 ∅ ∩ 1978). { } a , b = . Similarly, either N(x ) a, b = {or N(y}0) ∅ a, b = . ∩ { } ∅ Corollary 2. Every block graph is hereditary c- ∩ { } ∅ domishold. Every trivially perfect graph is heredi- Proof. If N(x) a0, b0 = and N(y) tary c-domishold. a0, b0 = , then∩ there { exists} an 6 (x,∅ y)-path in G ∩ {S, contrary} 6 ∅ to the fact that S is an (x, y)-separator.− Sketch of proof of Theorem 2. The implications The other statement follows similarly. (3) (1) (2) follow from Proposition 3. ⇒ ⇒ For the implication (2) (4), we only need 0 0 0 ⇒ Notice that C C = C Z = C Z = and to verify that none of the graphs in the set := so C C0 = C∩ Z = ∩C0 Z0 ∩= 4. Further,∅ C : k 4 F ,F H : i F1 is | ∪ | | ∪ | | ∪ | { k ≥ } ∪ { 1 2} ∪ { i ≥ } since every minimal separator in a chordal graph is ms-2-asummable. Let F . Suppose first that a clique (Dirac 1961), C and C0 are cliques. On the F is a cycle C for some k∈ F 4, let u , u , u , u 0 k ≥ 1 2 3 4 other hand, Z and Z are independent sets, there- be four consecutive vertices on the cycle. For a set fore C Z0 1 and C0 Z 1. S V (F ), let xS 0, 1 V (F ) denote the char- | ∩ | ≤ | ∩ | ≤ acteristic⊆ vector of ∈S, { defined} by xS = 1 if and Claim 2. N(C0) Z 1 and N(C) Z0 1. i | ∩ | ≤ | ∩ | ≤ only if i S. Let A = u1, u3 , B = u2, u4 , ∈ { } { } 0 0 C = u1, u2 and D = u3, u4 . Then, A and Proof. If N(C ) Z > 1 then Z N(C ). Since { } { } B are minimal separators of F , while C and D do C0 S = | , this implies∩ | that x and y⊆are in the same not contain any minimal separator of F . Therefore, connected∩ ∅ component of G S, a contradiction. xA and xB are true points of the minimal separator The other statement follows by− symmetry. C D function msF , while x and x are false points A B C D 0 of msF . Since x + x = x + x , the mini- Claim 2 implies that Z = Z . Up to symme- 6 mal separator function msF is 2-summable. If F try, it remains to analyze five cases, depending ∈ 0 0 F1,F2 Hi : i 1 , then let a and b be the whether the sets C,C ,Z,Z have vertices in com- two{ vertices} ∪ { of degree≥2 in}F , let N(a) = a , a , mon (where possible) or not. In what follows we { 1 2} N(b) = b1, b2 , let A = N(a), B = N(b), use the notation u v (resp. u  v) to de- { } ∼ C = a1, b1 and D = a2, b2 . The rest of the note the fact that two vertices u and v are adjacent proof{ is the same} as above.{ } (resp. non-adjacent). It remains to show the implication (4) ⇒ Case 1. C Z0 = Z Z0 = 1. (3). Since the class of F1,F2,H1,H2,... -free | ∩ | | ∩ | chordal graphs is hereditary,{ it is enough} to Without loss of generality, we may assume that a = x0. Since C Z = and a = x0 it follows show that every F1,F2,H1,H2,... -free chordal 0 ∩ ∅ 0 0 graph is ms-dually-Sperner.{ Suppose} for a contra- that x / Z, implying Z Z = y . Without loss of generality,∈ we may assume∩ that{ }0 . But diction that there exists a F ,F ,H ,H ,... - y = y 1 2 1 2 the fact that implies 0 0, leading to a free chordal graph G ={ (V,E) that is not} y a y x contradiction. ∼ ∼ ms-dually-Sperner. If G is disconnected, then HmS(G) = (V (G), ) and G is ms-dually- The case C0 Z = Z Z0 = 1 is symmetric Sperner. Thus, G is connected.{∅} Since G is not ms- to Case 1. | ∩ | | ∩ | 0 0 0 0 Case 2. Z Z = 1 and C Z = C Z = . F2 (otherwise). If y y , then, to avoid an in- | ∩ | ∩ ∩ ∅ ∼0 0 0 Without loss of generality, we may assume that duced C4 on y, a = x , a , y , we conclude that 0 0 0 0 0 0 { } x = x . Since a, b S and S separates x and y , y a . But now we have a copy of F1 induced we conclude that 6∈N(y0) a, b = . By sym- by∼x, b, a = x0, a0, y, y0 , a contradiction. Thus, 0 0 ∩ { } ∅ { 0 } 0 0 metry, N(y) a , b = , and consequently, y  y , implying also N(y) a , b = , since y0 S and y ∩S { 0. Since} S separates∅ x and y and otherwise the vertex set b, a ∩= {x0, y,} a0, b0∅, y0 in- 06∈ 0 0 6∈ 0 0 0 { 0 0 } a , b , y S = , we have N(y) a , b , y = duces either a copy of F1 (if N(y) a , b = 1) { } ∩ ∅ 0 ∩ { } | ∩{ }| , and, similarly, N(y ) a, b, y = . or of F2 (otherwise). ∅ ∩ { 0 } ∅ We must have y  y since otherwise G Since neither of the vertices a0, b0 and y0 is ad- contains either an induced C4 on the vertex set jacent to and is a clique containing , we con- 0 0 0 b S b y, a, a , y (if a a ) or an induced C5 on the clude that a0, b0, y0 S = . In particular, if K vertex{ set }y, a, x =∼x0, a0, y0 (otherwise). { } ∩ ∅ 0 { } denotes the component of G S containing a , this To avoid an induced copy of H1 on the vertex set implies 0 0 . By Lemma− 2, there exists a 0 0 0 0 b , y V (K) y, a, b, x = x , a , b , y , we may assume, with- vertex w V ∈(K) that dominates S. Since S sepa- { } 0 out loss of generality, that a a . rates x from∈ y, we have x, y K; without loss ∼0 0 * Suppose first that b b . Then also a b of generality, we may assume{ } that y / V (K). Let or 0 , since otherwise∼ 0 0 would∼ in- ∈ 0 0 0 a b a, a , b , b P = (w1, w2, . . . , wk) be a shortest a , b , y –w duce a∼ copy of . But now{ (depending} if we have 0 0 0 { } C4 path in K where w1 a , b , y and wk = w. one edge or both) the vertex set y, a, b, a0, b0, y0 ∈ { } { } Since w1 is not adjacent to b but wk is, we have induces a copy of either F1 or of F2. Therefore, k > 1. b b0.  Suppose that k = 2. If w y0, then the vertex Suppose that a0 b. But now, either the vertex  set b, a = x0, w, a0, b0, y0 induces either a copy set y, a, b, x = x∼0, a0, b0 induces a copy of F 2 of F{ (if N(w) a0, b0 } = 1) or of F (oth- (if a{ b0), or the vertex set} y, a, b, a0, b0, y0 in- 1 2  erwise), a| contradiction.∩ { Hence}| w y0. To avoid duces a copy of F (if a b0{). Therefore, a0 } b, 1  an induced C on the vertices w, a∼= x0, a0, y0 , and by symmetry, a b0∼. But now, the vertex set 4  we conclude that w a0. But{ now, the vertex set} y, a, b, x = x0, a0, b0 induces a copy of F , a con- 1 y, a = x0, b, w, a0,∼ y0 induces a copy of F , a tradiction.{ } 1 {contradiction. Therefore,} k 3. Case 3. C Z0 = C0 Z = 1 and Z Z0 = . ≥ | ∩ | | ∩ | ∩ ∅ To avoid an induced copy of a cycle of order Without loss of generality, we may assume that 0 0 0 at least 4, we conclude that vertex a = x dom- a = x and a = x. By Claim 1 it follows that 0 0 0 0 0 inates P . If y w2 then also a w2 and b  y . The fact that y  x = a implies y / S 0 ∼ ∼ 0 ∈ 0 b w2 (or otherwise we would have an induced (since S is a clique) and consequently also y  y ∼ 0 0 0 0 0 C4 on the vertex set a = x , w2, b , y or a = (otherwise x = a and y would be in the same 0 0 0 { } { 0 x , w2, a , y ) but that gives us an induced F1 on component of G S ). To avoid an (x, y)-path in } 0 0 0 − 0 the vertex set y , a , a = x , w2, w3, w4 (where G S, we conclude that y  b . Now, the vertices { 0 } − 0 0 0 0 w4 = b if k = 3). Therefore, y  w2. Without a = x , a = x, b, b , y, y induce either a copy 0 { 0 } loss of generality, we may assume that w1 = a . of F1 (if b  b ) or of F2 (otherwise). In either 0 0 0 But now, the vertex set y , a = w1, b , a = case, we reach a contradiction. 0 { 0 x , w2, w3 induces a copy of either F1 (if b  0 0 0 } Case 4. C Z = 1 and C Z = Z Z = . w2) or of F2 (otherwise), a contradiction. Without| loss∩ of| generality, we∩ may assume∩ that∅ a = x0. By Claim 2, we have N(C0) Z 1. The case C0 Z = 1 and C Z0 = Z Z0 = Thus, we may assume that x| / N(C∩0). Conse-| ≤ is symmetric| to∩ Case| 4. ∩ ∩ ∅ quently, N(x) a0, b0 = ∈and therefore also 0 x y0, for otherwise∩ { } we would∅ have a C in- It remains to consider the case when C Z =  4 0 0 ∩ duced by 0 0 . To avoid an 0 0 -path C Z = Z Z = . This case is analyzed de- a, a , y , x (x , y ) ∩ ∩ ∅ in G S0{, we conclude} that b y0. Moreover, pending on the number of edges between x, y  0 0 { } we also− have N(b) a0, b0 = , since otherwise and x , y , and on the number of edges between { } 0 0 the vertex set ∩ { 0 }0 ∅0 0 induces ei- a, b and a , b . Due to space constraints, we x, a = x , a , b, b , y { } { } ther a copy of{F (if N(b) a0, b0 } = 1) or of omit here the analysis of this case. 1 | ∩ { }| Discussion Berge, C. 1989. Hypergraphs, volume 45 of We conclude with a brief discussion on algorithmic North-Holland Mathematical Library. Amster- aspects of (hereditary) c-domishold graphs. The dam: North-Holland Publishing Co. Combina- characterization of hereditary c-domishold graphs torics of finite sets. in terms of forbidden induced subgraphs given by Butenko, S.; Kahruman-Anderoglu, S.; and Ursu- Theorem 2 can be used to develop a polynomial lenko, O. 2011. On connected domination in unit time recognition algorithm for this class. More- ball graphs. Optim. Lett. 5(2):195–205. over, since the set of all minimal separators of a Chandran, L. S., and Grandoni, F. 2006. A lin- given graph can be computed in output-polynomial ear time algorithm to list the minimal separators of time (Shen and Liang 1997), c-domishold graphs chordal graphs. Discrete Math. 306(3):351–358. can be recognized in polynomial time in any class Chandran, L. S.; Das, A.; Rajendraprasad, D.; and of graphs with only polynomially many minimal Varma, N. M. 2012. Rainbow connection number separators, by applying Proposition 2, and verify- and connected dominating sets. J. Graph Theory ing whether the minimal separator function ms G 71(2):206–218. is threshold (which can be done in polynomial time using the algorithm from (Peled and Sime- Chiarelli, N., and Milanic,ˇ M. 2013a. Lin- one 1985)). Since hereditary c-domishold graphs ear separation of total dominating sets in graphs. are chordal, and chordal graphs have a linear num- In Proc. 39th International Workshop on Graph- ber of minimal separators (Chandran and Grandoni Theoretic Concepts in Computer Science (WG 2006), the above approach can also be used to 2013), 165–176. compute a c-domishold structure of a given hered- Chiarelli, N., and Milanic,ˇ M. 2013b. To- itary c-domishold graph. tal domishold graphs: a generalization of thresh- We leave open the recognition problem for the old graphs, with connections to threshold hy- general case. pergraphs. Submitted. Preprint available on Problem 1. Determine the computational com- arXiv:1303.0944v2. plexity of recognizing c-domishold graphs. Chow, C. K. 1961. Boolean functions realizable with single threshold devices. Proc. IRE 49:370– Acknowledgments. The research presented in 371. this paper is partly co-financed by the European Chvatal,´ V., and Hammer, P. L. 1977. Aggrega- Union through the European Social Fund. Co- tion of inequalities in integer programming. In financing is carried out within the framework of Studies in integer programming (Proc. Workshop, the Operational Programme for Human Resources Bonn, 1975). Amsterdam: North-Holland. 145– Development for 2007-2013, 1. development pri- 162. Ann. of Discrete Math., Vol. 1. orities Promoting entrepreneurship and adaptabil- Crama, Y., and Hammer, P. L. 2011. Boolean func- ity; priorities 1. 3: Scholarship Scheme. The work tions, volume 142 of Encyclopedia of Mathemat- of the second author was supported in part by ics and its Applications. Cambridge: Cambridge Slovenian Research Agency, research program P1– University Press. Theory, algorithms, and applica- 0285 and research projects J1–4010,J1–4021,J1– tions. 5433, and N1–0011: GReGAS, supported in part by the European Science Foundation. Dirac, G. A. 1961. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg 25:71–76. 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