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Graphs with Dilworth Number Two are Pairwise Compatibility Graphs
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Electronic Notes in Discrete Mathematics 44 (2013) 31–38 www.elsevier.com/locate/endm
Graphs with Dilworth Number Two are Pairwise Compatibility Graphs T. Calamoneri and R. Petreschi Department of Computer Science, “Sapienza” University of Rome, Italy Email:[email protected],[email protected]
Abstract A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T , and two non-negative real numbers dmin and dmax , dmin ≤ dmax , such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w (u, v) ≤ dmax where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . When the constraints on the distance between the pairs of leaves concern only dmax or only dmin the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined. It is known that LP G ∩ mLP G is not empty and that threshold graphs are one of the classes contained in it. It is also known that threshold graphs are all the graphs with Dilworth number one, where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another. In this paper we do one step ahead toward the comprehension of the structure of the set LP G ∩ mLP G, proving that graphs with Dilworth number two belong to this intersection.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.006
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Introduction
A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T and two non-negative real numbers dmin and dmax , dmin ≤ dmax , such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w (u, v) ≤ dmax where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . In such a case, we say that G is a PCG of T for dmin and dmax ; in symbols, G = P CG(T, w, dmin , dmax ). The pairwise compatibility graph recognition problem consists in determining whether a given graph is PCG or not while the pairwise compatibility tree construction problem consists in finding out a tree T , an edge-weight function w and two suitable values, dmin and dmax , such that the given graph G is P CG(T, w, dmin , dmax ). Since 2003, when the PCG class was introduced by Kearney et al. [14] dealing with a sampling problem in a phylogenetic tree, this class of graphs has received great interest from researchers belonging to different fields, from computational biology to computational complexity and to graph theory. Indeed, the main application of the pairwise compatibility graph recognition problem remains in phylogenetics, in relation to the reconstruction of a tree expressing the evolutionary relations among organisms based on quantitative biological data [12]. Nevertheless, researchers interested in computational complexity theory are fascinated by these graphs because the clique problem is known to be polynomially solvable for PCGs [14] (while it is well known that, for a general graph, finding whether there is a clique of a given size is NP-complete [10]). Moreover, the class of PCGs appears interesting by itself from a graph theoretic point of view, since it is not known how to determine whether a graph G is PCG or not: nowadays, there are a few results proving that some special classes of graphs are PCGs [1,13,15,17,18] but only two sets of results for generic graphs, one affirming that all graphs with a number of vertices not greater than 7 are PCGs [4,16] and the other one proving that not all graphs are PCGs, by means of two graphs that cannot be a PCG, one with 15 vertices [18] and the other one with 8 vertices [8]. One of the most recent results concerned with this class of graphs [6] explores the relation between the PCG class and two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves deal only with dmax or only with dmin . More precisely, a graph G = (V, E) is called a leaf power graph, LPG (respectively minimum leaf power graph, mLPG) if there exists a tree T , a positive edge-weight function
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w on T , and a non-negative real number dmax (respectively dmin ) such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dT,w (u, v) ≤ dmax (respectively dT,w (u, v) ≥ dmin ), where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . In [6] the authors show that the union of these two proper subclasses does not coincide with the whole PCG class and that neither of the classes LPG and mLPG is contained in the other. Moreover, the class of threshold graphs is in both LPG and mLPG, together with other classes such as cliques – that are trivially in LP G ∩ mLP G. Threshold graphs correspond to graphs with Dilworth number one [7,9], where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another In this paper we consider the wider class of graphs with Dilworth number at most two and prove that it is also a subclass of LP G ∩ mLP G, so progressing toward the characterization of this intersection.
2
Preliminary definitions and properties
In this section we introduce some terminology and recall some definitions that will be used in the rest of the paper. The reader is referred to [2] for undefined terms and notation. We consider only simple and connected graphs G = (V, E) with vertex set V and edge set E. A graph G = (V, E) is a threshold graph if there is a positive real number S (the threshold) and for every vertex v there is a real weight a(v) such that (v, w) is an edge if and only if a(v) + a(w) ≥ S. Threshold graphs are completely determined by their set of vertices V , threshold S and vertexweight function a, so we indicate them as G = (V, a, S). Threshold graphs coincide with Dilworth one graphs [7], while graphs with Dilworth number at most two coincide with a generalization of threshold graphs, called threshold signed graphs [3]. A graph G = (V, E) is a threshold signed graph if there are two positive real numbers S, T (the thresholds) and for every vertex v there is a real weight a(v) < min(S, T ) such that (v, w) is an edge if and only if either |a(v) + a(w)| ≥ S or |a(v) − a(w)| ≥ T . In Figure 1.a) a threshold signed graph is depicted. Notice that if S = T then the threshold signed graph is simply a threshold graph. For any vertex v, if a(v) = 0 then v is an isolated vertex, so if (u, v) is an edge in a connected graph G, then a(u) · a(v) = 0. Consider an edge (u, v) of a threshold signed graph. It is not difficult to see that only one of the two conditions concerning the thresholds can be satisfied; so, if
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a(u) · a(v) > 0, meaning that a(u) and a(v) have the same signs, then it must be that |a(u) + a(v)| ≥ S and the edge (u, v) is called S-edge; if, on the contrary, a(u) · a(v) < 0, that is a(u) and a(v) have different sign, then it must be that |a(u) − a(v)| ≥ T and the edge is called T-edge. We call ES and ET the sets of S- and T -edges. Of course, E = ES ∪ ET . We can consider the partition of the vertices of G into two sets X and Y , V = X ∪ Y such that X = {x ∈ V s.t. a(x) < 0} and Y = {y ∈ V s.t. a(y) > 0}. As consequence, (u, v) is an S-edge (T-edge) if u and v are in the same (different) set X or Y . Finally, we remark that the graph GS = (V, ES ) is exactly the union of two vertex-disjoint threshold graphs, one corresponding to X and one to Y and, consequently, X and Y are two vertex-disjoint chains, while the graph GT = (V, ET ) is a bipartite graph [2]. From now on we will refer to a threshold signed graph as G = (V, a, S, T ) always assuming V = X ∪ Y and S > T , according to the construction presented in [3]. The class of threshold signed graphs is closed under complementation [3]. c c 0.5 d u u u u u u −1 −1 −2
u0.7 u1
u1.5
S = 2.5 T =2 a.
u1.8
0.7 1.5 1 1.8
0.7 1.5 1 1.8
u
u
u
u
u
dmin = 2.5 b.
u
u
1 1 2
u
u
u
u
dmin = 2.5 c.
Fig. 1. a) A threshold signed graph G; b) the star S4 that witnesses that the graph induced by the four vertices with positive weight (set Y in the text) is in mLPG; c) the caterpillar C that witnesses that G is in mLPG.
A caterpillar is a tree in which all the vertices are within distance one of a central path which is called the spine.
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Graphs with Dilworth Number Two are in LP G ∩ mLP G
In this section we show that any threshold signed graph is both a mLPG and a LPG exploiting a construction of the edge-weighted witness tree for threshold graphs. The known construction for threshold graphs [6] uses the definition – among the numerous equivalent ones of threshold graphs – based
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on the degree sequence; here we recall that construction by proposing a slight modification, in order to use the definition founded on the threshold S and the vertex-weight function a. Construction for Threshold Graphs. The pairwise compatibility tree of a threshold graph G = (V, a, S) is a star Sn with n + 1 vertices, a central vertex c and leaves v1 , . . . , vn ; the weight of each edge (c, vi ), w(c, vi ), is equal to a(vi ) (in Figure 1.b it is shown the star S4 corresponding to the threshold graph induced by the vertices with positive weights of the graph in Figure 1.a); it is easy to prove that G = mLP G(Sn , w, S). It is also possible to construct an analogous edge-weighted star showing that the threshold graph is in LPG, but we omit it because it is not necessary in the context of this section. 2 Now we are able to build a tree C that is a caterpillar, an edge-weight function w and two values dmin and dmax for showing that a threshold signed graph is both an mLPG and a LPG. Lemma 3.1 Let G = (V, a, S, T ), where V = X ∪ Y , be a threshold signed graph and S > T . A caterpillar C with edge-weight function w and a value dmin such that G = mLP G(C, w, dmin ) can be found in polynomial time. Proof. Let us focus on the two threshold graphs induced by Y and X. For them we construct two stars, S|Y | (whose central vertex is c) and S|X| (whose central vertex is d) using the construction given above. Notice that the vertexweight w(v) of each vertex v of the threshold graph induced by Y coincides with a(v). On the contrary, all the vertex-weights of the threshold graph induced by X are negative, so for each vertex v ∈ X we assign −a(v) to the edge incident to the leaf corresponding to v. In order to construct the caterpillar C, connect the central vertices c and d of stars S|X| and S|Y | by means of an edge with weight w(c, d) = S − T . Observe that this value is always positive, as S > T (see Figure 1.c). Now we have to prove that the mLPG generated by C with dmin = S is exactly the given threshold signed graph G. Consider any edge (u, v) of G. If u and v are both in X or both in Y , then |a(u) + a(v)| ≥ S and hence we have that the length of the edge-weighted path in C is dC (u, v) = w(x, u)+w(x, v) = |a(u) + a(v)| ≥ S, (with either x = c or x = d) so proving that (u, v) is also an edge of the mLPG. If u ∈ Y and v ∈ X then |a(u) − a(v)| ≥ T holds. Recalling that the vertices in X have negative weights while the vertices in Y have positive weights, the length of the corresponding path in C is dC (u, v) = w(u, c) + w(c, d) + w(d, v) = a(u) + S − T − a(v) = |(a(u) − a(v))| + S − T ≥ T + S − T = S hence (u, v) is also an edge of the mLPG. Let us now consider
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in G two not connected vertices u and v, then the following two inequalities hold: |a(u) + a(v)| < S and |a(u) − a(v)| < T . If u and v belong to the same set, their distance in C is dC (u, v) = w(x, u) + w(x, v) = |a(u) + a(v)| < S, (with either x = c or x = d), and hence (u, v) is not an edge of the mLPG. If, on the contrary, u ∈ Y and v ∈ X, then dC (u, v) = w(u, c) + S − T + w(d, v) = a(u) + S − T − a(v) = |(a(u) − a(v))| + S − T < T + S − T = S and hence, also in this case, (u, v) is not an edge of the mLPG. So the proof is concluded since we have proved that mLP G(C, w, S) contains all edges of G and only those. 2 Proposition 3.2 [5] The complement of every graph in LPG is in mLPG and conversely, the complement of every graph in mLPG is in LPG. In particular, if G = LP G(T, w, dmax ) then GC = mLP G(T, w, min{u,v}∈E(G) dT (u, v)); if G = mLP G(T, w, dmin ) then GC = LP G(T, w, max{u,v}∈E(G) dT,w (u, v)). Lemma 3.3 Let G be a threshold signed graph G = (V, a, S, T ), where V = X ∪ Y and S > T . There exists a caterpillar C with edge-weight function w and a value dmax such that G = LP G(C, w, dmax ). Proof. Consider the complement GC of G. As the class of threshold signed graphs is self-complemented, GC is also a threshold signed graph, and it is possible to determine its edge-weight function w and the two thresholds S and T , with S > T . Apply now the construction of Lemma 3.1 to find an edgeweighted caterpillar C and a value dmin so that GC = mLP G(C, w, dmin ). From Proposition 3.2, it follows that the graph complement of GC is G = LP G(C, w, dmax ), where dmax = max{u,v}∈E(G) dT,w (u, v). 2 Lemmas 3.1 and 3.3 easily imply the following result: Theorem 3.4 The class of graphs with Dilworth number at most two is in LP G ∩ mLP G.
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Two open problems
In this paper, we proved that graphs with Dilworth number at most two are in LP G ∩ mLP G. From this result, the following questions naturally arise: Is it possible to completely characterize the set LP G ∩ mLP G? Which graph classes are in the set? What can be said about the relation between graphs with Dilworth number at most 3 [11] and pairwise compatibility graphs?
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References [1] Brandst¨adt A.: On Leaf Powers. Technical report, University of Rostock, 2010. [2] Brandst¨ adt A., Le V. B. and Spinrad J.: Graph classes: a survey. SIAM Monographs on discrete mathematics and applications, 1999. [3] Benzaken C., Hammer P.L. and de Werra D.: Threshold characterization of graphs with Dilworth number 2. Journal of Graph Theory 9, 245–267, 1985. [4] Calamoneri T., Frascaria D. and Sinaimeri B.: All graphs with at most seven vertices are Pairwise Compatibility Graphs. The Computer Journal, 2012, to appear. doi:10.1093/comjnl/bxs087 [5] Calamoneri T., Montefusco E., Petreschi R. and Sinaimeri B.: Exploring Pairwise Compatibility Graphs. Theoretical Computer Science 468, 23–36, 2013. [6] Calamoneri T., Petreschi R. and Sinaimeri B.: On relaxing the constraints in pairwise compatibility graphs. In Proc. of WALCOM 2012, LNCS 7157,124– 135, Springer, Berlin, 2012. [7] Chvatal V. and Hammer P.L.: Aggregation of inequalities in integer programming. Annals of Discrete Math 1, 145–162, 1977. [8] Durocher S, Mondal D. and Rahman Md. S.: On Graphs that are not PCGs. In Proc. of WALCOM 2013, LNCS, 2013, to appear. [9] Foldes S. and Hammer P.L.: The Dilworth number of a graph. Annals of Discrete Math 2, 211–219, 1978. [10] Garey M. R. and Johnson D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979. [11] Hoang C.T. and Mahadev N.V.R.: A note on perfect orders. Discrete Math. 74, 77–84, 1989. [12] Jones N.C. and Pevzner P.A.: An Introduction to Bioinformatics Algorithms. MIT Press, Cambridge, 2004. [13] Kearney P.E. and Corneil D.G.: Tree powers. J. Algorithms 29(1), 111–131, 1998. [14] Kearney P. E., Munro J. I. and Phillips D.: Efficient generation of uniform samples from phylogenetic trees. In Proc. of WABI 2003, LNCS 2812, 177–189, Springer, Berlin, 2003.
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[15] Lin G.H., Jiang T. and Kearney P.E.: Phylogenetic k-root and Steiner k-root. In Proc. of ISAAC ’00, LNCS1969, 539–551, Springer, Berlin, 2000. [16] Phillips D.: Uniform sampling from phylogenetics trees. Masters Thesis, University of Waterloo, 2002. [17] Yanhaona M.N., Hossain K.S.M.T. and Rahman M.S.: Pairwise compatibility graphs. J. Appl. Math. Comput. 30, 479–503, 2009 . [18] Yanhaona M.N., Bayzid Md. S. and Rahman Md. S.: Discovering Pairwise compatibility graphs. Discrete Mathematics, Algorithms and Applications 2(4), 607–623, 2010 .
Electronic Notes in Discrete Mathematics 44 (2013) 31–38 www.elsevier.com/locate/endm
Graphs with Dilworth Number Two are Pairwise Compatibility Graphs T. Calamoneri and R. Petreschi Department of Computer Science, “Sapienza” University of Rome, Italy Email:[email protected],[email protected]
Abstract A graph G = (V, E) is called a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T , and two non-negative real numbers dmin and dmax , dmin ≤ dmax , such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w (u, v) ≤ dmax where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . When the constraints on the distance between the pairs of leaves concern only dmax or only dmin the two subclasses LPGs (Leaf Power Graphs) and mLPGs (minimum Leaf Power Graphs) are defined. It is known that LP G ∩ mLP G is not empty and that threshold graphs are one of the classes contained in it. It is also known that threshold graphs are all the graphs with Dilworth number one, where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another. In this paper we do one step ahead toward the comprehension of the structure of the set LP G ∩ mLP G, proving that graphs with Dilworth number two belong to this intersection.
1571-0653/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.endm.2013.10.006
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Introduction
A graph G = (V, E) is a pairwise compatibility graph (PCG) if there exists a tree T , a positive edge-weight function w on T and two non-negative real numbers dmin and dmax , dmin ≤ dmax , such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dmin ≤ dT,w (u, v) ≤ dmax where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . In such a case, we say that G is a PCG of T for dmin and dmax ; in symbols, G = P CG(T, w, dmin , dmax ). The pairwise compatibility graph recognition problem consists in determining whether a given graph is PCG or not while the pairwise compatibility tree construction problem consists in finding out a tree T , an edge-weight function w and two suitable values, dmin and dmax , such that the given graph G is P CG(T, w, dmin , dmax ). Since 2003, when the PCG class was introduced by Kearney et al. [14] dealing with a sampling problem in a phylogenetic tree, this class of graphs has received great interest from researchers belonging to different fields, from computational biology to computational complexity and to graph theory. Indeed, the main application of the pairwise compatibility graph recognition problem remains in phylogenetics, in relation to the reconstruction of a tree expressing the evolutionary relations among organisms based on quantitative biological data [12]. Nevertheless, researchers interested in computational complexity theory are fascinated by these graphs because the clique problem is known to be polynomially solvable for PCGs [14] (while it is well known that, for a general graph, finding whether there is a clique of a given size is NP-complete [10]). Moreover, the class of PCGs appears interesting by itself from a graph theoretic point of view, since it is not known how to determine whether a graph G is PCG or not: nowadays, there are a few results proving that some special classes of graphs are PCGs [1,13,15,17,18] but only two sets of results for generic graphs, one affirming that all graphs with a number of vertices not greater than 7 are PCGs [4,16] and the other one proving that not all graphs are PCGs, by means of two graphs that cannot be a PCG, one with 15 vertices [18] and the other one with 8 vertices [8]. One of the most recent results concerned with this class of graphs [6] explores the relation between the PCG class and two particular subclasses resulting from the cases where the constraints on the distance between the pairs of leaves deal only with dmax or only with dmin . More precisely, a graph G = (V, E) is called a leaf power graph, LPG (respectively minimum leaf power graph, mLPG) if there exists a tree T , a positive edge-weight function
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w on T , and a non-negative real number dmax (respectively dmin ) such that V coincides with the set of leaves of T , and there is an edge (u, v) ∈ E if and only if dT,w (u, v) ≤ dmax (respectively dT,w (u, v) ≥ dmin ), where dT,w (u, v) is the sum of the weights of the edges on the unique path from u to v in T . In [6] the authors show that the union of these two proper subclasses does not coincide with the whole PCG class and that neither of the classes LPG and mLPG is contained in the other. Moreover, the class of threshold graphs is in both LPG and mLPG, together with other classes such as cliques – that are trivially in LP G ∩ mLP G. Threshold graphs correspond to graphs with Dilworth number one [7,9], where the Dilworth number of a graph is the size of the largest subset of its vertices in which the close neighborhood of no vertex contains the neighborhood of another In this paper we consider the wider class of graphs with Dilworth number at most two and prove that it is also a subclass of LP G ∩ mLP G, so progressing toward the characterization of this intersection.
2
Preliminary definitions and properties
In this section we introduce some terminology and recall some definitions that will be used in the rest of the paper. The reader is referred to [2] for undefined terms and notation. We consider only simple and connected graphs G = (V, E) with vertex set V and edge set E. A graph G = (V, E) is a threshold graph if there is a positive real number S (the threshold) and for every vertex v there is a real weight a(v) such that (v, w) is an edge if and only if a(v) + a(w) ≥ S. Threshold graphs are completely determined by their set of vertices V , threshold S and vertexweight function a, so we indicate them as G = (V, a, S). Threshold graphs coincide with Dilworth one graphs [7], while graphs with Dilworth number at most two coincide with a generalization of threshold graphs, called threshold signed graphs [3]. A graph G = (V, E) is a threshold signed graph if there are two positive real numbers S, T (the thresholds) and for every vertex v there is a real weight a(v) < min(S, T ) such that (v, w) is an edge if and only if either |a(v) + a(w)| ≥ S or |a(v) − a(w)| ≥ T . In Figure 1.a) a threshold signed graph is depicted. Notice that if S = T then the threshold signed graph is simply a threshold graph. For any vertex v, if a(v) = 0 then v is an isolated vertex, so if (u, v) is an edge in a connected graph G, then a(u) · a(v) = 0. Consider an edge (u, v) of a threshold signed graph. It is not difficult to see that only one of the two conditions concerning the thresholds can be satisfied; so, if
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a(u) · a(v) > 0, meaning that a(u) and a(v) have the same signs, then it must be that |a(u) + a(v)| ≥ S and the edge (u, v) is called S-edge; if, on the contrary, a(u) · a(v) < 0, that is a(u) and a(v) have different sign, then it must be that |a(u) − a(v)| ≥ T and the edge is called T-edge. We call ES and ET the sets of S- and T -edges. Of course, E = ES ∪ ET . We can consider the partition of the vertices of G into two sets X and Y , V = X ∪ Y such that X = {x ∈ V s.t. a(x) < 0} and Y = {y ∈ V s.t. a(y) > 0}. As consequence, (u, v) is an S-edge (T-edge) if u and v are in the same (different) set X or Y . Finally, we remark that the graph GS = (V, ES ) is exactly the union of two vertex-disjoint threshold graphs, one corresponding to X and one to Y and, consequently, X and Y are two vertex-disjoint chains, while the graph GT = (V, ET ) is a bipartite graph [2]. From now on we will refer to a threshold signed graph as G = (V, a, S, T ) always assuming V = X ∪ Y and S > T , according to the construction presented in [3]. The class of threshold signed graphs is closed under complementation [3]. c c 0.5 d u u u u u u −1 −1 −2
u0.7 u1
u1.5
S = 2.5 T =2 a.
u1.8
0.7 1.5 1 1.8
0.7 1.5 1 1.8
u
u
u
u
u
dmin = 2.5 b.
u
u
1 1 2
u
u
u
u
dmin = 2.5 c.
Fig. 1. a) A threshold signed graph G; b) the star S4 that witnesses that the graph induced by the four vertices with positive weight (set Y in the text) is in mLPG; c) the caterpillar C that witnesses that G is in mLPG.
A caterpillar is a tree in which all the vertices are within distance one of a central path which is called the spine.
3
Graphs with Dilworth Number Two are in LP G ∩ mLP G
In this section we show that any threshold signed graph is both a mLPG and a LPG exploiting a construction of the edge-weighted witness tree for threshold graphs. The known construction for threshold graphs [6] uses the definition – among the numerous equivalent ones of threshold graphs – based
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on the degree sequence; here we recall that construction by proposing a slight modification, in order to use the definition founded on the threshold S and the vertex-weight function a. Construction for Threshold Graphs. The pairwise compatibility tree of a threshold graph G = (V, a, S) is a star Sn with n + 1 vertices, a central vertex c and leaves v1 , . . . , vn ; the weight of each edge (c, vi ), w(c, vi ), is equal to a(vi ) (in Figure 1.b it is shown the star S4 corresponding to the threshold graph induced by the vertices with positive weights of the graph in Figure 1.a); it is easy to prove that G = mLP G(Sn , w, S). It is also possible to construct an analogous edge-weighted star showing that the threshold graph is in LPG, but we omit it because it is not necessary in the context of this section. 2 Now we are able to build a tree C that is a caterpillar, an edge-weight function w and two values dmin and dmax for showing that a threshold signed graph is both an mLPG and a LPG. Lemma 3.1 Let G = (V, a, S, T ), where V = X ∪ Y , be a threshold signed graph and S > T . A caterpillar C with edge-weight function w and a value dmin such that G = mLP G(C, w, dmin ) can be found in polynomial time. Proof. Let us focus on the two threshold graphs induced by Y and X. For them we construct two stars, S|Y | (whose central vertex is c) and S|X| (whose central vertex is d) using the construction given above. Notice that the vertexweight w(v) of each vertex v of the threshold graph induced by Y coincides with a(v). On the contrary, all the vertex-weights of the threshold graph induced by X are negative, so for each vertex v ∈ X we assign −a(v) to the edge incident to the leaf corresponding to v. In order to construct the caterpillar C, connect the central vertices c and d of stars S|X| and S|Y | by means of an edge with weight w(c, d) = S − T . Observe that this value is always positive, as S > T (see Figure 1.c). Now we have to prove that the mLPG generated by C with dmin = S is exactly the given threshold signed graph G. Consider any edge (u, v) of G. If u and v are both in X or both in Y , then |a(u) + a(v)| ≥ S and hence we have that the length of the edge-weighted path in C is dC (u, v) = w(x, u)+w(x, v) = |a(u) + a(v)| ≥ S, (with either x = c or x = d) so proving that (u, v) is also an edge of the mLPG. If u ∈ Y and v ∈ X then |a(u) − a(v)| ≥ T holds. Recalling that the vertices in X have negative weights while the vertices in Y have positive weights, the length of the corresponding path in C is dC (u, v) = w(u, c) + w(c, d) + w(d, v) = a(u) + S − T − a(v) = |(a(u) − a(v))| + S − T ≥ T + S − T = S hence (u, v) is also an edge of the mLPG. Let us now consider
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in G two not connected vertices u and v, then the following two inequalities hold: |a(u) + a(v)| < S and |a(u) − a(v)| < T . If u and v belong to the same set, their distance in C is dC (u, v) = w(x, u) + w(x, v) = |a(u) + a(v)| < S, (with either x = c or x = d), and hence (u, v) is not an edge of the mLPG. If, on the contrary, u ∈ Y and v ∈ X, then dC (u, v) = w(u, c) + S − T + w(d, v) = a(u) + S − T − a(v) = |(a(u) − a(v))| + S − T < T + S − T = S and hence, also in this case, (u, v) is not an edge of the mLPG. So the proof is concluded since we have proved that mLP G(C, w, S) contains all edges of G and only those. 2 Proposition 3.2 [5] The complement of every graph in LPG is in mLPG and conversely, the complement of every graph in mLPG is in LPG. In particular, if G = LP G(T, w, dmax ) then GC = mLP G(T, w, min{u,v}∈E(G) dT (u, v)); if G = mLP G(T, w, dmin ) then GC = LP G(T, w, max{u,v}∈E(G) dT,w (u, v)). Lemma 3.3 Let G be a threshold signed graph G = (V, a, S, T ), where V = X ∪ Y and S > T . There exists a caterpillar C with edge-weight function w and a value dmax such that G = LP G(C, w, dmax ). Proof. Consider the complement GC of G. As the class of threshold signed graphs is self-complemented, GC is also a threshold signed graph, and it is possible to determine its edge-weight function w and the two thresholds S and T , with S > T . Apply now the construction of Lemma 3.1 to find an edgeweighted caterpillar C and a value dmin so that GC = mLP G(C, w, dmin ). From Proposition 3.2, it follows that the graph complement of GC is G = LP G(C, w, dmax ), where dmax = max{u,v}∈E(G) dT,w (u, v). 2 Lemmas 3.1 and 3.3 easily imply the following result: Theorem 3.4 The class of graphs with Dilworth number at most two is in LP G ∩ mLP G.
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Two open problems
In this paper, we proved that graphs with Dilworth number at most two are in LP G ∩ mLP G. From this result, the following questions naturally arise: Is it possible to completely characterize the set LP G ∩ mLP G? Which graph classes are in the set? What can be said about the relation between graphs with Dilworth number at most 3 [11] and pairwise compatibility graphs?
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