aa r X i v : . [ m a t h . C O ] O c t On chordal phylogeny graphs ∗ Soogang Eoh and Suh-Ryung Kim Department of Mathematics Education, Seoul National University, Seoul 08826mathfi[email protected], [email protected]
Abstract
An acyclic digraph each vertex of which has indegree at most i and outdegree atmost j is called an ( i, j ) digraph for some positive integers i and j . Lee et al. (2017)studied the phylogeny graphs of (2 ,
2) digraphs and gave sufficient conditions andnecessary conditions for (2 ,
2) digraphs having chordal phylogeny graphs. Theirwork was motivated by problems related to evidence propagation in a Bayesiannetwork for which it is useful to know which acyclic digraphs have their moralgraphs being chordal (phylogeny graphs are called moral graphs in Bayesian networktheory).In this paper, we extend their work. We completely characterize phylogenygraphs of (1 , i ) digraphs and ( i,
1) digraphs, respectively, for a positive integer i .Then, we study phylogeny graphs of a (2 , j ) digraphs, which is worthwhile in thecontext that a child has two biological parents in most species, to show that thephylogeny graph of a (2 , j ) digraph D is chordal if the underlying graph of D ischordal for any positive integer j . Especially, we show that as long as the underlyinggraph of a (2 ,
2) digraph is chordal, its phylogeny graph is not only chordal but alsoplanar.
Keywords: competition graph; moral graph; phylogeny graph; ( i, j ) digraph; chordalgraph; planar graph
Throughout this paper, we deal with simple digraphs. ∗ This research was supported by the National Research Foundation of Korea(NRF) funded by theKorea government(MEST) (No. NRF-2017R1E1A1A03070489) and by the Korea government(MSIP)(No. 2016R1A5A1008055). v v v v v v D v v v v v v v C ( D ) v v v v v v v P ( D )Figure 1: An acyclic digraph D , the competition graph C ( D ) of D , and the phylogenygraph P ( D ) of D .Given an acyclic digraph D , the competition graph of D , denoted by C ( D ), is the sim-ple graph having vertex set V ( D ) and edge set { uv | ( u, w ) , ( v, w ) ∈ A ( D ) for some w ∈ V ( D ) } . Since Cohen [2] introduced the notion of competition graphs in the study onpredator-prey concepts in ecological food webs, various variants of competition graphshave been introduced and studied.The notion of phylogeny graphs was introduced by Roberts and Sheng [8] as a vari-ant of competition graphs. Given an acyclic digraph D , the underlying graph of D ,denoted by U ( D ), is the simple graph with vertex set V ( D ) and edge set { uv | ( u, v ) ∈ A ( D ) or ( v, u ) ∈ A ( D ) } . The phylogeny graph of an acyclic digraph D , denoted by P ( D ),is the graph with the vertex set V ( D ) and edge set E ( U ( D )) ∪ E ( C ( D )).“Moral graphs” having arisen from studying Bayesian networks are the same as phy-logeny graphs. One of the best-known problems, in the context of Bayesian networks, isrelated to the propagation of evidence. It consists of the assignment of probabilities to thevalues of the rest of the variables, once the values of some variables are known. Cooper [3]showed that this problem is NP-hard. Most noteworthy algorithms for this problem aregiven by Pearl [7], Shachter [9] and by Lauritzen and Spiegelhalter [5]. Those algorithmsinclude a step of triangulating a moral graph, that is, adding edges to a moral graph toform a chordal graph.A hole of a graph is an induced cycle of length at least four. A graph is said to be chordal if it does not contain a hole. As triangulations of moral graphs play an importantrole in algorithms for propagation of evidence in a Bayesian network, studying chordalityof the phylogeny graphs of acyclic digraphs is meaningful.For positive integers i and j , an ( i, j ) digraph is an acyclic digraph such that eachvertex has indegree at most i and outdegree at most j . A graph G is a phylogeny graph (resp. an ( i, j ) phylogeny graph ) if there is an acyclic digraph D (resp. an ( i, j ) digraph D ) such that P ( D ) is isomorphic to G . Throughout this paper, we assume that variables2 and j belong to the set of positive integers unless otherwise stated.In this paper, we completely characterize the phylogeny graphs of (1 , j ) digraphs andthose of ( i,
1) digraphs (Theorem 2.1 and 2.8). Then we study the phylogeny graphs of(2 , j ) digraphs. We show that the phylogeny graph of any (2 , j ) digraph whose underlyinggraph is chordal is chordal (Theorem 3.6). Finally, we show that the phylogeny graph ofany (2 ,
2) digraph whose underlying graph is chordal is chordal and planar (Theorem 3.17). (1 , j ) phylogeny graphs and ( i, phylogeny graphs In this section, we characterize the (1 , j ) phylogeny graphs and the ( i,
1) phylogeny graphs.A component of a digraph D is the subdigraph of D induced by the vertex of acomponent of its underlying graph. Given an acyclic digraph D , it is easy to check that D ′ is a component of D if and only if P ( D ′ ) is a component of P ( D ). Thus,( ⋆ ) it is sufficient to consider only weakly connected digraphs (whose underlying graphsare connected) in studying phylogeny graphs of digraphs.First we take care of (1 , j ) phylogeny graphs.A vertex of degree one is called a pendant vertex .Given a graph G and a vertex v of G , we denote the set of neighbors of v in G by N G ( v ). We call N G ( v ) ∪ { v } the closed neighborhood of v and denote it by N G [ v ]. We call∆( G ) := max {| N G ( v ) | | v ∈ V ( G ) } the maximum degree of G . Theorem 2.1.
For a positive integer j , a graph is a (1 , j ) phylogeny graph if and only ifit is a forest with the maximum degree at most j + 1 .Proof. By ( ⋆ ), it suffices to show that, for a positive integer j , a connected graph is a(1 , j ) phylogeny graph if and only if it is a tree with the maximum degree at most j + 1.To show the “only if” part, suppose that a connected graph G is a (1 , j ) phylogeny graphfor some positive integer j . Then there is a (1 , j ) digraph D such that P ( D ) is isomorphicto G . Since every vertex of D has indegree at most one, P ( D ) = U ( D ). Since P ( D )is connected, U ( D ) is connected. Moreover, since D is a (1 , j ) digraph, U ( D ) has themaximum degree at most j + 1. If U ( D ) contained a cycle C , then there would exist avertex on C of indegree at least two by the acyclicity of D . Therefore U ( D ) does notcontain a cycle, and so U ( D ) is tree.Now we show the “if” part. If T is a tree with one or two vertices, then it is obviously a(1 ,
1) phylogeny graph. We take a tree T with at least three vertices and let j = ∆( T ) − u . We regard T as a rooted tree with the root u and define an oriented tree −→ T , which is acyclic, with V ( −→ T ) = V ( T ) as follows. We take an edge xy in T . Then d T ( u, x ) = d T ( u, y ) + 1 or d T ( u, y ) = d T ( u, x ) + 1. If the former, ( y, x ) ∈ A ( −→ T ) and if the latter, ( x, y ) ∈ A ( −→ T ).By definition, U ( −→ T ) = T . Moreover, u has indegree zero and outdegree one, and each3ertex in −→ T except u has indegree one in −→ T . Then, since the degree of each vertex in T is at most j + 1, the outdegree of each vertex in −→ T is at most j . Therefore −→ T is a (1 , j )digraph. Since each vertex in −→ T has indegree at most one, P ( −→ T ) = U ( −→ T ) = T .If P ( D ) is triangle-free for an acyclic digraph D , then the indegree of each vertex isat most one in D , for otherwise, the vertex with indegree at least two form a trianglewith two in-neighbors in P ( D ). Thus, the following corollary immediately follows fromTheorem 2.1. Corollary 2.2.
For any positive integers i and j , if an ( i, j ) phylogeny graph is triangle-free, then it is a forest with the maximum degree at most j + 1 . A clique of a graph G is a set X of vertices of G with the property that every pair ofdistinct vertices in X are adjacent in G . A maximal clique of a graph G is a clique X ofvertices of G , such that there is no clique of G that is a proper superset of X . The sizeof a maximum clique of a graph G is called a clique number and denoted by ω ( G ).Given a digraph D with n vertices, a one-to-one correspondence f : V ( D ) → [ n ] iscalled an acyclic labeling of D if f ( u ) > f ( v ) for any arc ( u, v ) in D . It is well-knownthat D is acyclic if and only if there is an acyclic labeling of D .Given a digraph D and a vertex v of D , we denote the set of in-neighbors of v in D by N − D ( v ). We call N − D ( v ) ∪ { v } the closed in-neighborhood of v and denote it by N − D [ v ]. Inaddition, for a vertex v in D , we call | N + D ( v ) | and | N − D ( v ) | the outdegree and the indegree of v , respectively, and denote it by d + D ( v ) and d − D ( v ), respectively.Given a graph G , a vertex v of G is called a simplicial vertex if N G [ v ] forms a cliquein G .Now we consider phylogeny graphs of ( i,
1) digraphs.
Lemma 2.3.
Let D be a nontrivial weakly connected ( i, digraph for some positiveinteger i and f be an acyclic labeling of D . Then every maximal clique in P ( D ) is in theform of the closed in-neighborhood of the vertex with the least f -value among the verticesin the maximal clique.Proof. Let X be a maximal clique in P ( D ) and x be the vertex having the least f -valueamong the vertices in X . Suppose X N − D [ x ]. Then there is a vertex y ∈ X such that( y, x ) / ∈ A ( D ). Since x has the least f -value among the vertices in X , ( x, y ) / ∈ A ( D ). Yet,since x and y are adjacent in P ( D ), they have a common out-neighbor, say z , in D . Bythe hypothesis that D is an ( i,
1) digraph, z is the only out-neighbor of x and y . Since x has the least f -value among the vertices in X , z / ∈ X . Since N − D [ z ] forms a clique in P ( D ), X N − D [ z ] by the maximality of X . That is, there exists a vertex w in X but notin N − D [ z ]. Then w = z . Since z is the unique out-neighbor of x and y in D , ( x, w ) / ∈ A ( D )and ( y, w ) / ∈ A ( D ). Furthermore, since w / ∈ N − D [ z ], neither w and x nor w and y have acommon out-neighbor in D . However, w , x , and y belong to X , so ( w, x ) and ( w, y ) arearcs in D , which is a contradiction to the hypothesis that D is a ( i,
1) digraph. Hence X ⊂ N − D [ x ]. Since N − D [ x ] is a clique in P ( D ), X = N − D [ x ] by the maximality of X .4 emma 2.4. Given a nontrivial weakly connected ( i, digraph D for a positive integer i , the set of all the maximal cliques in P ( D ) is exactly the set { N − D [ u ] | u ∈ V ( D ) and d − D ( u ) ≥ } . Proof.
Let f be an acyclic labeling of D . Take a maximal clique Y in P ( D ). ByLemma 2.3, Y = N − D [ y ] for the vertex y having the least f -value among the verticesin Y . Since D is nontrivial and weakly connected, | Y | ≥ y has an in-neighbor in D , i.e. d − D ( y ) ≥ u of indegree at least onein D . Let v be an in-neighbor of u in D . Suppose, to the contrary, N − D [ u ] is not maximal.Then there is a maximal clique X properly containing N − D [ u ]. By Lemma 2.3, X = N − D [ x ]for the vertex x with the least f -value among the vertices in X . Since N − D [ u ] is properlycontained in N − D [ x ], u = x . In addition, since N − D [ u ] is included in N − D [ x ], v is also anin-neighbor of x in D . Thus the outdegree of v is at least two, which contradicts the factthat D is an ( i,
1) digraph. Therefore N − D [ u ] forms a maximal clique in P ( D ) and thiscompletes the proof.A diamond is a graph obtained from K by deleting an edge. A graph is called diamond-free if it does not contain a diamond as an induced subgraph. Lemma 2.5.
The phylogeny graph of a weakly connected ( i, digraph for a positiveinteger i is diamond-free and chordal.Proof. Let D be a weakly connected ( i,
1) digraph for a positive integer i . We prove thelemma statement by induction on | V ( D ) | . If | V ( D ) | = 1 or 2, then the statement istrivially true. Suppose that | V ( D ) | = n + 1 and the lemma statement is true for anyweakly connected ( i,
1) digraph with n vertices ( n ≥ D is acyclic, there is avertex u of indegree zero in D . Since D is a weakly connected ( i,
1) digraph, d + D ( u ) = 1.Thus there is a unique out-neighbor v of u in D . Then, as u has indegree of zero in D , wemay conclude that, for a vertex w in D , u is adjacent to w in P ( D ) if and only if w = v or w is an in-neighbor of v in D , i.e. N P ( D ) [ u ] = N − D [ v ] . (1)Since the indegree and the outdegree of u are zero and one, respectively, D − u is weaklyconnected. Obviously D − u is an ( i,
1) digraph. Thus, by the induction hypothesis, P ( D − u ) is diamond-free and chordal. Take two vertices x and y in V ( D ) \ { u } . Since u has indegree zero, u cannot be a common out-neighbor of x and y . Therefore, x and y are adjacent in P ( D ) − u if and only if ( x, y ) ∈ A ( D ) or ( y, x ) ∈ A ( D ) or they have acommon out-neighbor other than u in D if and only if x and y are adjacent in P ( D − u ).Thus we have shown that P ( D ) − u = P ( D − u ). By (1), u is simplicial in P ( D ), so P ( D )is chordal. Now it remains to show that P ( D ) is diamond-free.5uppose that P ( D ) has a diamond. Then, since P ( D ) − u is diamond-free, everydiamond of P ( D ) contains u and a vertex which is not adjacent to u in P ( D ). Let z be a vertex on a diamond which is not adjacent to u in P ( D ). Then z is not containedin N − D [ v ] \ { u } and is adjacent to two vertices y and y in N − D [ v ] \ { u } by (1). Since P ( D ) − u = P ( D − u ), z is adjacent to y and y in P ( D − u ). Moreover, u is not apendant vertex, so v has an in-neighbor distinct from u in D . Then v has indegree atleast one in D − u , so N − D − u [ v ] is a maximal clique in P ( D − u ) by Lemma 2.4. Obviously N − D − u [ v ] = N − D [ v ] \ { u } , so N − D [ v ] \ { u } is a maximal clique in P ( D − u ). Then, since z belongs to P ( D − u ) and is not contained in N − D [ v ] \ { u } , there exist a vertex w in N − D − u [ v ]which is not adjacent to z in P ( D − u ). Then the subgraph induced by z , w , y , and y is a diamond in P ( D − u ) and we have reached a contradiction. Lemma 2.6.
Let D be an ( i, digraph for a positive integer i and f be an acyclic labelingof D . Suppose that non-disjoint vertex sets X and Y form distinct maximal cliques in P ( D ) , respectively. Then X and Y have exactly one common vertex, namely v , and f ( v ) = min { f ( w ) | w ∈ X } or min { f ( w ) | w ∈ Y } whereas f ( v ) > min { f ( w ) | w ∈ X ∪ Y } . Proof.
By ( ⋆ ), we may assume that U ( D ) is connected. By Lemma 2.5, P ( D ) is diamond-free, so | X ∩ Y | ≤
1. Then, by the hypothesis that X and Y are non-disjoint vertex sets, | X ∩ Y | = 1. Let v be the vertex common to X and Y . Since X and Y form maximalcliques, X = N − D [ x ] and Y = N − D [ y ] for the vertices x and y with the smallest f -valuesamong the vertices in X and the vertices in Y , respectively, by Lemma 2.3. Since X and Y are distinct, x = y . If v / ∈ { x, y } , then x and y are two distinct out-neighbors of v ,which is impossible. Thus v ∈ { x, y } . Without loss of generality, we may assume v = x .Since x and y have the the smallest f -values among the vertices in X and the verticesin Y , respectively, v has the least f -value among the vertices in X but not among thevertices in Y , and the lemma statement is true.We shall completely characterize the ( i,
1) phylogeny graphs in terms of “clique graph”which was introduced by Hamelink [4].
Definition 2.7.
The clique graph of a graph G , denoted by K ( G ), is a simple graph suchthat • every vertex of K ( G ) represents a maximal clique of G ; • two vertices of K ( G ) are adjacent when they share at least one vertex in commonin G . Theorem 2.8.
For some positive integer i , a graph G is an ( i, phylogeny graph if andonly if it is a diamond-free chordal graph with ω ( G ) ≤ i + 1 and its clique graph is a forest. roof. By ( ⋆ ), it is sufficient to show that a connected graph G is an ( i,
1) phylogenygraph for some positive integer i if and only if it is a diamond-free chordal graph with ω ( G ) ≤ i + 1 and its clique graph is a tree. To show the “only if” part, suppose thata connected graph G is an ( i,
1) phylogeny graph for some positive integer i . Then G = P ( D ) for some weakly connected ( i,
1) digraph D . By Lemma 2.3, ω ( G ) ≤ i + 1.In addition, by Lemma 2.5, P ( D ) is diamond-free and chordal. Now we show that theclique graph K ( G ) is a tree. As the clique graph of a connected graph is connected, it issufficient to show that K ( G ) is acyclic. Suppose, to the contrary, that K ( G ) contains acycle C := X X · · · X r X for an integer r ≥ X , . . . , X r of G . Let f be an acyclic labeling of D . We denote by x i the vertex which has the least f -valuein X i for each i = 1, 2, . . . , r . By Lemma 2.6, X ∩ X = { x } or X ∩ X = { x } .Without loss of generality, we may assume that X ∩ X = { x } so that f ( x ) < f ( x ).By Lemma 2.6 again, X ∩ X = { x } or X ∩ X = { x } . Suppose that X ∩ X = { x } .Then f ( x ) < f ( x ) and x ∈ X ∩ X . By Lemma 2.6, X ∩ X = { x } , and either f ( x ) = f ( x ) or f ( x ) = f ( x ), which contradicts the fact that f ( x ) < f ( x ) and f ( x ) < f ( x ). Thus X ∩ X = { x } . Continuing in this way, we may show that X i ∩ X i +1 = { x i +1 } for each i ∈ { , , . . . , r − } and X r ∩ X = { x } . By Lemma 2.3, X i = N − D [ x i ] for each i ∈ { , , . . . , r } . Therefore ( x , x r ) ∈ A ( D ) and ( x i +1 , x i ) ∈ A ( D )for each i ∈ { , , . . . , r − } . Thus x → x r → · · · → x → x is a directed cycle in D and we reach a contradiction to the acyclicity of D . Hence K ( G ) does not contain a cycleand so the “only if” part is true.To show the “if” part, suppose that a connected graph G is diamond-free and chordalwith ω ( G ) ≤ i + 1 and that K ( G ) is a tree for some positive integer i . If G is a completegraph, then it has at most i + 1 vertices and is obviously an ( i,
1) phylogeny graph. Thuswe may assume that G is not a complete graph. Then K ( G ) is not a trivial tree. Weshow by induction on | V ( G ) | that G is an ( i,
1) phylogeny graph. Since G is connectedand not complete, | V ( G ) | ≥
3. If | V ( G ) | = 3, then G is a path of length two and, byTheorem 2.1, a (1 ,
1) phylogeny graph. Assume that a connected non-complete graph isan ( i,
1) phylogeny graph if it is a diamond-free chordal graph with less than n verticesand the cliques of size at most i + 1 and its clique graph is a tree for n ≥
4. Supposethat | V ( G ) | = n . Since K ( G ) is not a trivial tree, it contains a pendant vertex. Let X be a pendant vertex and Y be the neighbor of X in K ( G ). Then | X ∩ Y | ≥
1. Since G is a connected graph with at least four vertices, by the maximality of X and Y , 2 ≤ | X | and 2 ≤ | Y | . By Lemma 2.6, X ∩ Y = { u } for some vertex u . Since | X | ≥
2, there existsa vertex v in X \ { u } . Since K ( G ) does not contain a triangle, X and Y are the onlymaximal cliques that contain u in G and so( † ) G − v does not have a maximal clique containing u other than X \{ v } (not necessarilymaximal) and Y .Furthermore, since X is a pendant vertex in K ( G ), every vertex in X \ { u } is a simplicialvertex in G and therefore v is a simplicial vertex of G . Then the closed neighborhood of v G is X . Moreover, it is obvious that G − v is a connected diamond-free chordal graphwith ω ( G − v ) ≤ i +1 and K ( G − v ) is a tree. Therefore, by the induction hypothesis, G − v is an ( i,
1) phylogeny graph. Thus there is an ( i,
1) digraph D ∗ such that P ( D ∗ ) = G − v .Let f ∗ be an acyclic labeling of D ∗ . Case 1 . The vertex u has the least f ∗ -value in Y . Then, by Lemma 2.3, Y = N − D ∗ [ u ].Consider the case in which u has no out-neighbor in D ∗ . Then, by Lemma 2.6, X \ { v } = { u } . Adding the vertex v and the arc ( u, v ) to D ∗ results in an ( i,
1) digraph whosephylogeny graph is G . Now consider the case in which u has an out-neighbor w in D ∗ .Then f ∗ ( w ) < f ∗ ( u ) and d − D ∗ ( w ) ≥
1. Since d − D ∗ ( w ) ≥ N − D ∗ [ w ] forms a maximal cliqueby Lemma 2.4. Since f ∗ ( w ) < f ∗ ( u ) and f ∗ ( u ) is the minimum in Y , N − D ∗ [ w ] is distinctfrom Y . Since N − D ∗ [ w ] contains u , N − D ∗ [ w ] = X \{ v } by ( † ). Since | X | ≤ i +1, | X \{ v }| ≤ i and so d − D ∗ ( w ) ≤ i −
1. Adding the vertex v and the arc ( v, w ) to D ∗ results in an ( i, G . Case 2 . The vertex u does not have the least f ∗ -value in Y . Then u has the least f ∗ -value in X \ { v } by Lemma 2.6. Thus, if u has no in-neighbor in D ∗ , then X \ { v } = { u } ,and so adding the vertex v and the arc ( v, u ) to D ∗ results in an ( i,
1) digraph whosephylogeny graph is G . Now consider the case in which u has an in-neighbor w in D ∗ .Then d − D ∗ ( u ) ≥
1, so N − D ∗ [ u ] forms a maximal clique by Lemma 2.4. By Lemma 2.3, u has the least f ∗ -value in N − D ∗ [ u ]. Since u does not have the least f ∗ -value in Y , N − D ∗ [ u ] isdistinct from Y . Since N − D ∗ [ u ] contains u , N − D ∗ [ u ] = X \ { v } by ( † ). Since | X | ≤ i + 1, | X \ { v }| ≤ i and so d − D ∗ ( u ) ≤ i −
1. Adding the vertex v and the arc ( v, u ) to D ∗ resultsin an ( i,
1) digraph whose phylogeny graph is G .The union of two graphs G and H is the graph having its vertex set V ( G ) ∪ V ( H ) andedge set E ( G ) ∪ E ( H ). If V ( G ) ∩ V ( H ) = ∅ , we refer to their union as a disjoint union. Proposition 2.9.
For a graph G , the following statements are equivalent.(i) G is a (1 , j ) phylogeny graph and an ( i, phylogeny graph for some positive integers i and j ;(ii) G is a disjoint union of paths;(iii) G is a (1 , phylogeny graph.Proof. By Theorems 2.1 and 2.8, it is immediately true that (ii) is equivalent (iii). Obvi-ously, (iii) implies (i). Now we show that (i) implies (ii). By Theorem 2.1, G is a forest.If G has a vertex of degree at least three, then K ( G ) contains a triangle as each edge in G is a maximal clique, which contradicts Theorem 2.8. Therefore each vertex in G hasdegree at most two and so G is a disjoint union of paths. Remark 2.10.
Theorems 2.1 and 2.8 tell us that an ( i, j ) phylogeny graph for positiveintegers i and j with i = 1 or j = 1 is diamond-free and chordal.8 (2 , j ) phylogeny graphs In this section, we focus on phylogeny graphs of (2 , j ) digraphs for a positive integer j .We thought that it is worth studying them in the context that a child has two biologicalparents in most species.For an acyclic digraph D , an edge is called a cared edge in P ( D ) if the edge belongsto the competition graph C ( D ) but not to the U ( D ). For a cared edge xy ∈ P ( D ), thereis a common out-neighbor v of x and y and it is said that xy is taken care of by v or that v takes care of xy . A vertex in D is called a caring vertex if an edge of P ( D ) is takencare of by the vertex.For example, the edges v v , v v , v v , v v , and v v of P ( D ) in Figure 1 are carededges and the vertices v , v , v , v , and v are vertices taking care of v v , v v , v v , v v , and v v , respectively. Proposition 3.1.
Suppose that the phylogeny graph of a (2 , j ) digraph D contains a hole H for a positive integer j . Then no vertex on H takes care of an edge on H .Proof. Suppose, to the contrary, that there exists a vertex v on H which takes care of anedge xy on H . Then { x, y, v } forms a triangle in P ( D ), so yv or vx is a chord of H in P ( D ) and we reach a contradiction.Given a (2 , j ) digraph D , suppose that P ( D ) has a hole H and e , e , . . . , e t are thecared edges on H . Let w , w , . . . , w t be vertices taking care of e , e , . . . , e t , respectively.Since the indegree of w i is at most two in D for i = 1 , . . . , t , w , w , . . . , w t are distinct.We let W = { w , w , . . . , w t } and call W a set extending H by extending the notionintroduced in Lee et al. [6]. By Proposition 3.1, W ⊂ V ( D ) \ V ( H ) . (2)Therefore we may obtain a cycle in U ( D ) from H by replacing each edge e i with a pathof length two from one end of e i to the other end of e i with the interior vertex w i . We callsuch a cycle the cycle obtained from H by W . Let L be the subgraph of U ( D ) inducedby V ( H ) ∪ W . We call L the subgraph of U ( D ) obtained from H by W . These notionsextend the ones introduced in Lee et al. [6].Lee et al. [6] showed that, for a (2 ,
2) digraph D such that the holes of P ( D ) aremutually vertex-disjoint and no hole in U ( D ) has length 4 or 6, the number of holes in U ( D ) is greater than or equal to the number of holes in P ( D ). Theorem 3.2 ([6]) . Let H be a hole of the phylogeny graph P ( D ) of a (2 , digraph D .Then there is a hole φ ( H ) in the underlying graph U ( D ) of D such that • φ ( H ) equals H if H is a hole in U ( D ) ; • φ ( H ) is a hole in U ( D ) only containing vertices in the subgraph obtained from H by a set extending H otherwise. oreover, if the holes of P ( D ) are mutually vertex-disjoint and no hole in U ( D ) haslength or , then there exists an injective map from the set of holes in P ( D ) to the setof holes in U ( D ) . We shall devote the first part of this section to extending the above theorem given in[6]. To do so, we need the following lemmas.
Lemma 3.3 ([1]) . Given a graph G and a cycle C of G with length at least four, supposethat a section Q of C forms an induced path of G and contains a path P with length atleast two none of whose internal vertices is incident to a chord of C in G . Then P canbe extended to a hole H in G so that V ( P ) ( V ( H ) ⊂ V ( C ) and H contains a vertex on C not on Q . Lemma 3.4.
Let D be a (2 , j ) digraph and f be an acyclic labeling of D for a positiveinteger j . In addition, let H be a hole of P ( D ) , W be a set extending H , and w be avertex with the least f -value in V ( H ) ∪ W . Then w ∈ W . Moreover, there is a hole φ ( H ) in U ( D ) such that w ∈ V ( φ ( H )) and V ( φ ( H )) ⊂ V ( H ) ∪ W .Proof. Let H = u u · · · u l u for an integer l ≥
4. To reach a contradiction, we supposethat w ∈ V ( H ). Without loss of generality, we may assume that w = u . Supposethat u u and u u l are edges of U ( D ). Then, since u has the least f -value in V ( H ),( u , u ) ∈ A ( D ) and ( u l , u ) ∈ A ( D ) and so { u , u , u l } forms a triangle in P ( D ), whichis a contradiction to the supposition that H is a hole in P ( D ). Therefore u u or u u l isa cared edge in P ( D ). Without loss of generality, we may assume u u is a cared edgein P ( D ). Then u and u have a common out-neighbor, say v , in W , which implies that f ( v ) < f ( u ). Thus we have reached a contradiction and so w ∈ W .Now we show that the “moreover” part of the lemma statement is true. Let C be thecycle in U ( D ) obtained from H by W . Without loss of generality, we may assume that u u l is taken care of by w . Then ( u , w ) ∈ A ( D ) and ( u l , w ) ∈ A ( D ).Suppose, to the contrary, that C has a chord which is incident to w in U ( D ). Let xw be a chord of C in U ( D ). Then x / ∈ { u , u l } . Moreover, since w has the least f -valuein V ( H ) ∪ W , ( w, x ) / ∈ A ( D ). Then u , u l , and x are in-neighbors of w in D , whichcontradicts the hypothesis that D is a (2 , j ) digraph. Hence there is no chord of C whichis incident to w in U ( D ). Since u u l is a cared edge in P ( D ), u wu l is an induced pathin U ( D ). By applying Lemma 3.3 for P = Q = u wu l , we may conclude that “moreover”part of the lemma statement is true.Now we are ready to extend Theorem 3.2 to not only make it valid for (2 , j ) digraphsbut also strengthen it. Theorem 3.5.
For a positive integer j , let H be a hole of the phylogeny graph P ( D ) of a (2 , j ) digraph D . Then there is a hole in U ( D ) which only contains vertices in thesubgraph of U ( D ) obtained from H by a set extending H . Moreover, if P ( D ) has a holeand the holes of P ( D ) are mutually edge-disjoint, then there exists an injective map fromthe set of holes in P ( D ) to the set of holes in U ( D ) . roof. The first part of this theorem is immediately true by Lemma 3.4.To show the second part of the theorem statement, we assume that P ( D ) has a holeand the holes in P ( D ) are mutually edge-disjoint. Let f be an acyclic labeling of D , { H , . . . , H l } be the set of holes in P ( D ), and W i be a set extending H i for each i = 1 , . . . , l .Let w i be the vertex with the least f -value in V ( H i ) ∪ W i for each i = 1 , . . . , l . Then,by Lemma 3.4, w i ∈ W i and there exists a hole φ ( H i ) such that w i ∈ V ( φ ( H i )) and V ( φ ( H i )) ⊂ V ( H i ) ∪ W i for each i = 1 , . . . , l . At this point, we may regard φ as a mapfrom the set of the holes in P ( D ) to the set of holes in U ( D ).In the following, we show that φ is injective. Suppose, to the contrary, that φ ( H j ) = φ ( H k ) for some j and k satisfying 1 ≤ j < k ≤ l . Since w i is the vertex with the least f -value in V ( H i ) ∪ W i and V ( φ ( H i )) ⊂ V ( H i ) ∪ W i , w i has the least f -value in V ( φ ( H i ))for each i ∈ { j, k } . Then, since φ ( H j ) = φ ( H k ), w j = w k and so w j ∈ W j ∩ W k . Thus w j has two in-neighbors on H j and two in-neighbors on H k in D . Then, by the hypothesisthat H j and H k are edge-disjoint, w j has at least three distinct in-neighbors in D , whichviolates the indegree restriction on D . Hence φ ( H j ) = φ ( H k ) for any j and k satisfying1 ≤ j < k ≤ l and we have shown that φ is injective.The underlying graph of an ( i, j ) digraph D being chordal does not guarantee that thephylogeny graph of D is chordal. For example, the underlying graph of the (3 ,
2) digraphgiven in Figure 1 is chordal whereas its phylogeny graph has a hole v v v v v . However,if i ≤ j = 1, then it does guarantee by the above theorem together with Theorems 2.1and 2.8. As a matter of fact, we have shown the following theorem. Theorem 3.6.
Let D ∗ i,j be the set of ( i, j ) digraphs whose underlying graphs are chordalfor positive integers i and j . Then the phylogeny graph of D is chordal for any D ∈ D ∗ i,j if and only if i ≤ or j = 1 . By Theorem 3.6, the phylogeny graph of a (2 , j ) digraph D is chordal if the underlyinggraph of D is chordal for any positive integer j . By the way, if j = 2, then the underlyinggraph being chordal guarantees not only P ( D ) being chordal but also P ( D ) being planar,which will be to be shown later in this section. By the way, Lee et al. [6] showed that a(2 ,
2) phylogeny graph is K -free. Theorem 3.7 ([6]) . For any (2 , digraph D , the phylogeny graph of D is K -free. We shall extend this theorem in two aspects. On one hand, we find a sharp upperbound for the clique number of (2 , j ) phylogeny graph for any positive integer j . On theother hand, we show that the phylogeny graph P ( D ) of a (2 ,
2) digraph D is planar if theunderlying graph of D is chordal by showing that P ( D ) is K -minor-free and K , -minor-free.For a positive integer k , a graph G is k -degenerate if any subgraph of G contains avertex having at most k neighbors in it. Lemma 3.8.
For a positive integer j , every (2 , j ) phylogeny graph is ( j + 2) -degenerate. v v v v v v v Figure 2: A (2 ,
1) digraph and (2 ,
2) digraph whose phylogeny graphs contain K and K ,respectively. Proof.
Let D be a (2 , j ) digraph for a positive integer j and f be an acyclic labeling of D . We take a subgraph H of P ( D ) and the vertex u which has the least f -value in V ( H ).Then the out-neighbors of u in D cannot be in V ( H ). Thus an edge incident to u in H is either a cared edge or the edge in U ( D ) corresponding to an arc incoming toward u in D . Since u has at most j out-neighbors and each of the out-neighbors has at most onein-neighbor other than u in D , there are at most j cared edges which are incident to u in H . Moreover, since u has at most two in-neighbors in D , there are at most two edgesincident to u in H which correspond to arcs incoming toward u in D . Thus u has degreeat most j + 2 in H . Since H was arbitrarily chosen, P ( D ) is ( j + 2)-degenerate.The following theorem gives a sharp upper bound for the clique number of (2 , j )phylogeny graph for any positive integer j to extend the Theorem 3.7. Theorem 3.9.
Let D be a (2 , j ) digraph for a positive integer j . Then ω ( P ( D )) ≤ ( j + 2 if j ≤ j + 3 otherwise;and the inequalities are tight.Proof. It is known that if a graph G is k -degenerate, then ω ( G ) ≤ k + 1. Thus, byLemma 3.8, ω ( P ( D )) ≤ j + 3. By Theorems 2.8 and 3.7, ω ( P ( D )) ≤ j + 2 if j ≤ j ≤ j ≥
3, we construct a (2 , j ) digraph in the following way. Westart with an empty digraph D with vertex set { v , . . . v j +3 } . We add to D the vertices a , , . . . , a ,j +1 and the arcs ( v , a ,i ), ( v i , a ,i ) for i = 2, . . . , j + 1 and arcs ( v j +2 , v ),( v j +3 , v ) to obtain a digraph D . Then D is a (2 , j )-digraph with every vertex except v having outdegree at most one and E := { v j +2 v j +3 } ∪ { v v i | i = 2 , . . . , j + 3 } is an edge set of P ( D ). We add to D the vertices a , , . . . , a ,j − , a ,j +1 , a ,j +2 and thearcs ( v , a ,i ), ( v i , a ,i ) for each i ∈ [ j + 2] \ { , , j } and arcs ( v j , v ), ( v j +3 , v ) to obtain12 digraph D . Then D is a (2 , j )-digraph with every vertex except v and v havingoutdegree at most two and E := E ∪ { v j v j +3 } ∪ { v v i | i = 3 , . . . , j + 3 } is an edge set of P ( D ).For each ℓ ∈ [ j − \ { , } , we add to D ℓ − the vertices a ℓ,ℓ +1 , . . . , a ℓ,j +1 and the arcs( v ℓ , a ℓ,i ), ( v i , a ℓ,i ) for i = ℓ + 1, . . . , j + 1 and arcs ( v j +2 , v ℓ ), ( v j +3 , v ℓ ) to obtain a digraph D ℓ . Then, for each ℓ ∈ [ j − \ { , } is a (2 , j )-digraph with every vertex except v , . . . ,and v l having outdegree at most ℓ and E ℓ := E ℓ − ∪ { v ℓ v i | i = ℓ + 1 , . . . , j + 3 } is an edge set of P ( D ℓ ). Therefore v i is adjacent to each of v , . . . , v j +3 except itself for i = 1, . . . , j −
1. Now we add to D j − the arcs ( v j +3 , v j +1 ), ( v j +2 , v j ), and ( v j +1 , v j ) toobtain a (2 , j ) digraph D j . Clearly, v j , . . . , v j +3 are mutually adjacent in P ( D j ) (recallthat the edges v j +2 v j +3 and v j v j +3 are contained in E and E , respectively). Thus v , . . . , v j +3 form a clique of size j + 3 in P ( D j ).From Theorems 2.1 and 2.8, we know that the clique number of a (1 , j ) phylogenygraph is at most two and the clique number of an ( i,
1) phylogeny graph is at most i + 1for any positive integers i and j .In the rest of this section, we shall show that the phylogeny graph P ( D ) of a (2 , D is planar if the underlying graph of D is chordal.The following lemma is a known fact. Lemma 3.10.
The class of chordal graphs is closed under contraction.
We denote by G · e the graph obtained by contracting a graph G by an edge e in G . Lemma 3.11.
For a graph G and two adjacent vertices u and v in G , let K be a cliquewith at least three vertices in G · uv . If z is the vertex in K obtained by identifying u and v , then one of the following is true: • K \ { z } ⊂ N G ( u ) ; • K \ { z } ⊂ N G ( v ) ; • the subgraph of G induced by ( K \ { z } ) ∪ { u, v } contains a hole in G .Proof. Suppose that K \ { z } 6⊂ N G ( u ) and K \ { z } 6⊂ N G ( v ). Then there is a vertex w and x in K \ { z } such that w is not adjacent to u and x is not adjacent to v . Since K isa clique in G · uv , w and x are adjacent to v and u , respectively, in G , and so uxwvu is ahole in G . 13 emma 3.12. A chordal graph G is K ω ( G )+1 -minor-free.Proof. Denote ω ( G ) by ω for simplicity’s sake. Suppose, to the contrary, that G contains K ω +1 as a minor. Then, since K ω +1 is complete, G contains an induced subgraph H such that K ω +1 is obtained from H by only contraction. Moreover, we may regard H asan induced subgraph of G for which the smallest number of contractions are required toobtain K ω +1 . Then, since G is chordal, H is also chordal. Clearly H is K ω +1 -free, so atleast one edge of H is contracted to obtain K ω +1 . Let uv be the last edge contracted toobtain K ω +1 from H . Let L be the second last graph obtained in the series of contractionsto obtain K ω +1 from H , that is, L · uv = K ω +1 . Then, by Lemma 3.11, V ( L ) \ { u, v } ⊂ N L ( u ) or V ( L ) \ { u, v } ⊂ N L ( v ) or L contains a hole. If V ( L ) \ { u, v } ⊂ N L ( u ) or V ( L ) \ { u, v } ⊂ N L ( v ), then L − v or L − u is isomorphic to K ω +1 , which contradictsthe choice of H . Thus V ( L ) \ { u, v } 6⊂ N L ( u ) and V ( L ) \ { u, v } 6⊂ N L ( v ), and so L contains a hole. However, since H is chordal, by Lemma 3.10, L is chordal and we reacha contradiction. Theorem 3.13.
For a positive integer j and a (2 , j ) digraph, if its underlying graph ischordal, then its phylogeny graph is K j +3 -minor-free if j ≤ and K j +4 -minor-free if j ≥ .Proof. Let D be a (2 , j ) digraph for a positive integer j whose underlying graph is chordal.Then, by Theorem 3.6, P ( D ) is chordal. Moreover, ω ( P ( D )) ≤ ( j + 2 , if j ≤ j + 3 , otherwise . by Theorem 3.9. Thus P ( D ) is K j +3 -free (resp. K j +4 -free) if j ≤ j ≥ P ( D ) is K j +3 -minor-free (resp. K j +4 -minor-free) if j ≤ j ≥ , j ) digraph whose underlying graph is non-chordal(see Figure 3). Corollary 3.14.
If the underlying graph of a (2 , digraph is chordal, then its phylogenygraph is K -minor-free. In the following, we show that the phylogeny graph of (2 ,
2) digraph whose underlyinggraph is chordal is K , -minor-free.The join of two graphs G and G is denoted by G ∨ G and has the vertex set V ( G ) ∪ V ( G ) and the edge set E ( G ) ∪ E ( G ) ∪ { xy | x ∈ G and y ∈ G } . Let I n denote a set of n isolated vertices in a graph for a positive integer n . Lemma 3.15.
For any (2 , digraphs, if its underlying graph is chordal, then its phy-logeny graph is K ∨ I -minor-free. v v v v v v v v D v v v v v v v v v P ( D )Figure 3: A (2 ,
2) digraph D whose phylogeny graph contains K as a minor. Proof.
Let G be the phylogeny graph of a (2 ,
2) digraph D whose underlying graph ischordal. Then, by Theorem 3.6 and Corollary 3.14, G is chordal and K -minor-free.Suppose, to the contrary, that K ∨ I is a minor of G . Then G contains a subgraph H such that either H = K ∨ I or K ∨ I is obtained from H by using edge deletions orcontractions. Let f be an acyclic labeling of D .Suppose that H = K ∨ I . If H is not an induced subgraph of G , then two vertices of I are adjacent in G , and so K is a subgraph of G , which is impossible. Thus H is an inducedsubgraph of G . We denote the vertices of K in H by x , x , x and the vertices of I in H by y , y , y . We may assume that f ( x ) < f ( x ) < f ( x ) and f ( y ) < f ( y ) < f ( y ).If f ( x ) < f ( y ), then the outdegree of x in the subdigraph D H of D induced by V ( H )is zero, which implies d H ( x ) ≤ D is a (2 ,
2) digraph), a contradiction. Thus f ( y ) < f ( x ) < f ( x ) < f ( x ) and f ( y ) < f ( y ) < f ( y ) . (3)If x has two in-neighbors in D H , then they must be y and y , which implies their beingadjacent in G , a contradiction. Therefore x has at most one in-neighbor in D H . Since D is a (2 ,
2) digraph and d H ( x ) = 5, x has exactly one in-neighbor and two out-neighborsin D H , and two cared edges in H are incident to x . The in-neighbor of x in D H is y or y by (3).Let y be the in-neighbor of x in D H . Then y ∈ { y , y } and f ( y ) > f ( x ). Thus, by(3), none of x , x , and y is an in-neighbor of y in D H . Since D is a (2 ,
2) digraph, y has at most one out-neighbor other than x in D H . Then, since d H ( y ) = 3, by (3), one of x y and x y is a cared edge in G taken care of by x or x . Since f ( x ) < f ( x ), x y is acared edge in G taken care of by x .Let v be a vertex joined to x by a cared edge in H . Then x and v have a commonout-neighbor in D . Since x has all of its two out-neighbors in D H , the common out-neighbors of x and v should be in H . Since there are two cared edges incident to x in15 , the two out-neighbors of x take care of those two cared edges incident to x . Since y has the least f -value among the vertices in H , y cannot be none of the other ends of twocared edges incident to x in G . Hence y must be one of the two out-neighbors of x in D H which takes care of a cared edge incident to x . Since { y , y , y } is an independentset in G , neither y nor y can be an in-neighbor of y in D H . Thus x or x is the vertexjoined to x which is taken care of by y in D H .If x is an in-neighbor of y in D H , then x y x yx is a hole in U ( D ) since { y, y } ⊂ I and x x is a cared edge in G which is not an edge in U ( D ). Thus x is an in-neighborof y in D H . In the following, we shall claim that x x y x yx is a hole in U ( D ) to reacha contradiction. Since { y, y } ⊂ I , y and y are not adjacent in U ( D ). Since x x is acared edge in G , x and x are not adjacent in U ( D ). If x x is an edge of U ( D ), thenthere is an arc ( x , x ) since f ( x ) < f ( x ), which contradicts the indegree condition on x . Therefore x and x are not adjacent in U ( D ). By applying a similar argument, wemay show that neither x and y nor y and x are adjacent in U ( D ).Thus H = K ∨ I and so K ∨ I is obtained from H by using edge deletions orcontractions. Then, K ∨ I may be obtained from the subgraph of G induced by V ( H )by using edge deletions or contractions, so we may assume that H as an induced subgraphof G . Then H is chordal. If an edge deletion was required to obtain K ∨ I from H , thenit would mean that G contains K as a minor, which is impossible. Thus, we may assumethat K ∨ I is obtained from H by only contractions.Let H ∗ be a graph obtained from H by applying the smallest number of contractionsto contain K ∨ I as a subgraph. Since H is chordal, H ∗ is chordal by Lemma 3.10.Let x , x , x be the vertices of K and y , y , y be the vertices of I for K ∨ I contained in H ∗ . Let H ′ be the graph to which the last contraction is applied in theprocess of obtaining H ∗ and e = uv be the edge contracted lastly. Then H ′ is chordalby Lemma 3.10. By the choice of H ∗ , u and v are identified to become a vertex in { x , x , x , y , y , y } . Case 1 . The vertices u and v are identified to become one of y , y , y . Without lossof generality, we may assume that u and v are identified to become the vertex y . ByLemma 3.11, { x , x , x } ⊂ N H ′ ( u ) or { x , x , x } ⊂ N H ′ ( v ) or { x , x , x , u, v } containsa hole in H ′ . Since H ′ is chordal, { x , x , x } ⊂ N H ′ ( u ) or { x , x , x } ⊂ N H ′ ( v ). Then { x , x , x , y , y , u } or { x , x , x , y , y , v } forms K ∨ I in H ′ , which contradicts thechoice of H ∗ . Case 2 . The vertices u and v are identified to become one of x , x , x . Then each of y , y , y is adjacent to one of u , v in H ′ . Without loss of generality, we may assume that u and v are identified to become the vertex x . By Lemma 3.11, { x , x } ⊂ N H ′ ( u ) or { x , x } ⊂ N H ′ ( v ) or { x , x , u, v } contains a hole in H ′ . Since H ′ is chordal, { x , x } ⊂ N H ′ ( u ) or { x , x } ⊂ N H ′ ( v ). Without loss of generality, we may assume that { x , x } ⊂ N H ′ ( u ).If u is adjacent to each of y , y , y , then { x , x , u, y , y , y } forms K ∨ I in H ′ , acontradiction to the choice of H ∗ . Thus u is not adjacent to one of y , y , y in H ′ . Withoutloss of generality, we may assume that u is not adjacent to y in H ′ . Then v is adjacent to16 in H ′ . If v is not adjacent to one of x and x , then x y vux or x y vux is a hole in H ′ and we reach a contradiction. Thus v is adjacent to x and x . If one of y , y is adjacentto both of u and v , then x , x , u , and v together with it form K in H ′ , a contradiction.Therefore { N H ′ ( u ) ∩ { y , y , y } , N H ′ ( v ) ∩ { y , y , y }} is a partition of { y , y , y } . Thus | N H ′ ( u ) ∩ { y , y , y }| + | N H ′ ( v ) ∩ { y , y , y }| = 3. Without loss of generality, we mayassume that | N H ′ ( u ) ∩{ y , y , y }| = 1. Then { x , x , v, y , y , y , u }\ ( N H ′ ( u ) ∩{ y , y , y } )forms K ∨ I in H ′ and we reach a contradiction. Theorem 3.16.
For any (2 , digraph D , if the underlying graph of D is chordal, thenthe phylogeny graph of D is K , -minor-free.Proof. Suppose, to the contrary, that K , is a minor of P ( D ). Then K , is obtained from P ( D ) by edge deletions or vertex deletions or contractions. Let ( X, Y ) be a bipartitionof K , . Among the edge deletions, the vertex deletions, and the contractions to obtain K , from P ( D ), we only take all the vertex deletions and all the contractions and applythem in the same order as the order in which vertex deletions and contractions appliedto obtain K , from P ( D ). Let H ∗ be a graph obtained from P ( D ) in this way. Then H ∗ contains K , as a spanning subgraph. In addition, since P ( D ) is chordal, H ∗ is chordalby Lemma 3.10 (it is clear that the chordality is preserved under vertex deletions). Ifthere is a pair of nonadjacent vertices in H ∗ in each of X and Y , then those four verticesform a hole in H ∗ and we reach a contradiction. Thus X or Y forms a clique in H ∗ andso H ∗ contains K ∨ I as a spanning subgraph. Then K ∨ I is a minor of P ( D ), whichcontradicts Lemma 3.15. Hence P ( D ) is K , -minor-free. Theorem 3.17.
For any (2 , digraphs, if its underlying graph is chordal, then its phy-logeny graph is chordal and planar.Proof. Let D be a (2 ,
2) digraph whose underlying graph is chordal. Then, by Theo-rem 3.6, P ( D ) is chordal. Furthermore, by Corollary 3.14 and Theorem 3.16, P ( D ) isplanar. Corollary 3.18.
A chordal graph one of whose orientations is a (2 , digraph is planar.Proof. Let G be a chordal graph one of whose orientations, namely D , is a (2 ,
2) digraph.Then U ( D ) is G which is chordal. Thus, by Theorem 3.17, P ( D ) is planar. Since U ( D )is a subgraph of P ( D ), U ( D ) is planar. References [1] Jihoon Choi, Soogang Eoh, and Suh-Ryung Kim. A new minimal chordal completion. arXiv preprint arXiv:1810.05280 , 2018.[2] Joel E Cohen. Interval graphs and food webs: a finding and a problem.
RANDCorporation Document , 17696, 1968. 173] Gregory F Cooper. The computational complexity of probabilistic inference usingbayesian belief networks.
Artificial intelligence , 42(2-3):393–405, 1990.[4] Ronald C Hamelink. A partial characterization of clique graphs.
Journal of Combi-natorial Theory , 5(2):192–197, 1968.[5] Steffen L Lauritzen and David J Spiegelhalter. Local computations with probabilitieson graphical structures and their application to expert systems.
Journal of the RoyalStatistical Society. Series B (Methodological) , pages 157–224, 1988.[6] Seung Chul Lee, Jihoon Choi, Suh-Ryung Kim, and Yoshio Sano. On the phylogenygraphs of degree-bounded digraphs.
Discrete Applied Mathematics , 233:83–93, 2017.[7] Judea Pearl. Fusion, propagation, and structuring in belief networks.
Artificial intel-ligence , 29(3):241–288, 1986.[8] Fred S Roberts and Li Sheng. Phylogeny graphs of arbitrary digraphs.
MathematicalHierarchies in Biology , pages 233–238, 1997.[9] Ross D Shachter. Probabilistic inference and influence diagrams.