L-Graphs and Monotone L-Graphs
Abu Reyan Ahmed, Felice De Luca, Sabin Devkota, Alon Efrat, Md Iqbal Hossain, Stephen Kobourov, Jixian Li, Sammi Abida Salma, Eric Welch
LL -Graphs and Monotone L -Graphs Abu Reyan Ahmed, Felice De Luca, Sabin Devkota, Alon Efrat, Md Iqbal Hossain,Stephen Kobourov, Jixian Li, Sammi Abida Salma, and Eric Welch
Department of Computer ScienceUniversity of Arizona
Abstract. An L -segment consists of a horizontal and a vertical straight line whichform an L . In an L -embedding of a graph, each vertex is represented by an L -segment, and two segments intersect each other if and only if the correspond-ing vertices are adjacent in the graph. If the corner of each L -segment in an L -embedding lies on a straight line, we call it a monotone L -embedding. In thispaper we give a full characterization of monotone L -embeddings by introduc-ing a new class of graphs which we call “non-jumping" graphs. We show that agraph admits a monotone L -embedding if and only if the graph is a non-jumpinggraph. Further, we show that outerplanar graphs, convex bipartite graphs, intervalgraphs, 3-leaf power graphs, and complete graphs are subclasses of non-jumpinggraphs. Finally, we show that distance-hereditary graphs and k -leaf power graphs( k ≤ ) admit L -embeddings. Geometric representations of graphs have been used to reveal intriguing connectionsbetween the continuous world of geometry and the discrete world of combinatorialstructures. Having a geometric representation is much more than just a way to displaya graph, as it reveals underlying structures that can often be described only using ge-ometry. A good geometric representation of a graph also leads to algorithmic solutionsfor purely graph-theoretic questions that, on the surface, do not seem to have anythingto do with geometry. Examples of this include rubber band representations in planaritytesting [20], circle-contact representations in balanced graph partitioning and approxi-mating optimal bisection [29], volume-respecting embeddings in approximation algo-rithms for graph bandwidth [15], and orthogonal representations in algorithms for graphconnectivity and graph coloring [21].In an intersection representation of a graph, vertices are geometric objects (e.g.,curves) and edges are realized by intersections (e.g., curve crossings). Among the mostgeneral types of intersection graphs are string-graphs , or graphs that admit a stringrepresentation , in which vertices are represented by arbitrary curves in the plane; seeFig. 1(a-b). String-graphs find a practical application in the modeling of integrated thinfilm RC circuits, where some pairs of conductors in a circuit can cross [28]. The classof k -string-graphs contains the graphs that have a string representation with at most k intersections between two strings, where k ≥ . Not every graph is a string-graph; forinstance, the full subdivision graph of the graph K does not have a string representa-tion; see Fig. 1(f). a r X i v : . [ c s . C G ] M a r Ahmed et al.
Fig. 1. (a) A graph G , (b) a string representation of G , and (c) a B -VPG representation of G . (d)an L -embedding of G , (e) a monotone L -embedding of G , and (f) a graph that is not an L -graph . Planar graphs are known to be -string graphs [8,9,14]. Chalopin and Gonçalvesstrengthen this result by proving a conjecture of Scheinerman [27] that every planargraph has a segment representation [10], where the segments have arbitrary slopes andintersect at arbitrary angles. The class of segment ( SEG ) graphs is included in the classof -string-graphs. The recognition of string-graphs is NP-hard [19,22].Another widely-studied class of graphs is the Vertex Path Grid ( VPG ) class, intro-duced by Asinowski et al. [1,2]. The class of k -Bends VPG ( B k -VPG ) graphs restrictsthe number of bends of the orthogonal paths to k , with k ≥ ; see Fig. 1(c). The classof B k -VPG-graphs is equivalent to the class of string-graphs [1,2]. Chaplick et al. [11]showed that for every fixed k , the recognition of B k -VPG-graph is NP-complete evenwhen the input graph is given by a B k +1 -VPG representation. The B k -VPG represen-tation is related to the edge intersection graphs of paths in a grid ( EPG-graphs ) intro-duced by Golumbic et al. [18]. In an EPG representation, the vertices are representedas paths on a grid, and two vertices are adjacent if and only if their corresponding pathsshare a grid edge. Pergel and Rz ˛a˙zewski [25] proved that is NP-complete to recognize -bend-EPG-graphs.The study of B k -VPG graphs is motivated by practical applications in circuit lay-outs [7,23]. n the knock-knee layout model, the layout may have multiple layers, and oneach layer, the vertex intersection graph of paths on a grid is an independent set. Thiscorresponds to a graph coloring problem, and the minimum coloring problem of VPG-graphs defines the knock-knee multiple layout with minimum number of layers. Thismodel is used by Asinowski et al. [2], who studied VPG-graphs and showed that intervalgraphs and trees are both subfamilies of B -VPG, and that circle graphs are contained inthe class B -VPG (where circle graphs are string graphs in which the strings are chordsof a circle). Since the problem of coloring a circle graph is NP-complete [17], it followsthat the coloring problem is also NP-complete for B -VPG-graphs. Asinowski et al. [2]proved that the coloring problem remains NP-complete even for B -VPG-graphs.The class B k -VPG contains all planar graphs, and a central question is how small k can be. Asinowski et al. [2] showed that every planar graph is a B -VPG-graph andChaplick and Ueckerdt [12] showed that every planar graph is a B -VPG-graph.In B -VPG-graphs, four possible L -shapes, L , L , L and L , are may be used to repre-sent vertices. In an L -graph, the vertices are represented with only one of these L -shapes.Biedl and Derka [5,6] show that series-parallel graphs, Halin-graphs, and outerplanargraphs are L -graphs. Felsner et al. [16] show that every planar -tree is an L -graph, and -Graphs and Monotone L -Graphs 3 that full subdivisions of planar graphs and line graphs of planar graphs are L -graphs.On the other hand, full subdivisions of non-planar graphs are not L -graphs [28].In the rest of this paper we restrict our focus to graphs that have an L -representation,which we refer to as L -graphs. Formally, an L -segment consists of a horizontal and avertical straight line which form an L . An L -embedding is a drawing of G in which eachvertex is drawn as an L -segment, and two segments intersect each other if and only ifthe corresponding vertices are adjacent in the graph. G is an L -graph if it admits an L -embedding . If the corner of each L -segment in an L -embedding lies on a straight line,then it is called a monotone L -embedding . A graph is called a monotone L -graph if itadmits a monotone L -embedding . Our contributions:
We study L -graphs and monotone L -graphs and summarize ourresults as follows: – We introduce a new class of graphs which we call “non-jumping graph" and a newvertex labeling which we call “non-jumping labeling." – We give a full characterization of monotone L -graphs by showing that a graph ad-mits a monotone L -embedding if and only if the graph is a non-jumping graph. – We show that given a graph G on n vertices and m edges with labeling γ , there is an O ( n log n + m ) time algorithm to determine whether γ is a non-jumping labeling. – We show that outerplanar graphs, convex bipartite graphs, interval graphs, and com-plete graphs are subclasses of non-jumping graphs. – We show that distance-hereditary graphs and k -leaf power graphs ( k ≤ ) admit L -embeddings.The rest of the paper is organized as follows. Section 2 defines some preliminarygraph-theoretic terminology. In Section 3, we define a “non-jumping graph” and showthat (bull, dart, gem)-free chordal graphs, interval graphs, outerplanar graphs, completegraphs, and convex bipartite graphs are non-jumping graphs. We also provide an algo-rithm to compute a monotone L -embedding of a non-jumping graph, and describe someof the properties of non-jumping graphs. In Section 4, we show that distance-hereditarygraphs and 4-leaf power graphs admit L -embeddings. We conclude the paper with someopen problems. In this section we introduce several definitions. For graph-theoretic definitions not de-scribed here, see [24].Let G = ( V, E ) be a graph with a set of vertices V and a set of edges E . We saythat G is connected if there is a path between every pair of vertices in V . A cycle of G is a path in which every vertex is reachable from itself. G is a tree if it does not containany cycles. G is planar if it can be embedded in the plane without edge crossings, and outerplanar if it has a planar drawing in which all vertices of G are placed on the outerface of the drawing. G is bipartite if its vertices can be partitioned into sets R and B such that every edge connects a vertex in R to one in B . The set of neighbors of v isdenoted by N ( v ) . If a bijective mapping f : B → { , , . . . , | B |} exists such that for all r ∈ R , and for any two vertices x, y ∈ N ( r ) , there does not exist a vertex z ∈ B \ N ( r ) Ahmed et al. such that f ( x ) < f ( z ) < f ( y ) , then G is called a convex bipartite graph. G is called an interval graph , and a set of intervals S is called an interval representation of G , if thereexists a one-to-one correspondence between vertices of G and intervals in S , such that u and v are adjacent in G , if and only if, their corresponding intervals intersect.Let G (cid:48) = ( V (cid:48) , E (cid:48) ) be a graph such that V (cid:48) ⊆ V and E (cid:48) ⊆ E . Then G (cid:48) is called a subgraph of G . The subgraph G (cid:48) is an induced subgraph of G if E (cid:48) consists of all theedges in E that have both endpoints in V (cid:48) . G is a distance-hereditary graph if and onlyif for every pair of vertices u , v all induced path between u and v have the same length. G is a k -leaf power graph if there is a tree T whose leaves correspond to the verticesof G in such a way that two vertices are adjacent in G precisely when their distance in T is at most k . We say that G is a leaf power graph if it is a k -leaf power for some k .A vertex u ∈ V is a pendant vertex if it has degree 1. For two vertices u, v ∈ V , if u and v are neighbors and N ( u ) \ { v } = N ( v ) \ { u } , then u and v are called true twins .If u and v are not neighbors and N ( u ) = N ( v ) , we say that u and v are false twins .A vertex v is called simplicial in G if the subgraph of G induced by the vertex set { v } ∪ N ( v ) is a complete graph. An ordering of { v , v , . . . , v n } is a perfect elimi-nation ordering of G if each v i is simplicial in the subgraph induced by the vertices { v , v , . . . , v i } . G is a chordal graph if it has a perfect elimination ordering.An L -segment consists of a horizontal and a vertical straight-line segment whichtogether look exactly like an L , with no rotation. Let L be an L -embedding of G and v be a vertex of G . We denote the corresponding L -segment of v in L by L ( v ). The L -segment is defined by its corner position, the height of its vertical and the width ofits horizontal line segments, denoted by ( v.x, v.y ) , v.h , and v.w , respectively. Let L ( u )and L ( v ) be two L -segments in L . The segments L ( u ) and L ( v ) might cross each othermultiple times in case of overlapping horizontal or vertical segments. In this paper weconsider only L -embeddings with single crossings, so that if ( u, v ) ∈ E then either thehorizontal segment of L ( v ) crosses the vertical segment of L ( u ), or the vertical segmentof L ( v ) crosses the horizontal segment of L ( u ). In this section we give a formal definition of a non-jumping graph. Then we showthat several classes of graphs are non-jumping graphs. Before we define non-jumpinggraphs, we must first define a non-jumping labeling of a graph G = ( V, E ) . A non-jumping labeling of G is a vertex labeling v , v , . . . , v n such that if ( v i , v k ) , ( v j , v l ) ∈ E and i < j < k < l , then ( v j , v k ) ∈ E . Figure 2(a) provides an example of anon-jumping labeling. If G admits a non-jumping labeling, then we say that G is a non-jumping graph ; if G has no non-jumping labeling then G is called a jumping graph . Ifa vertex labeling contains a vertex v j such that ( v i , v k ) , ( v j , v l ) ∈ E but ( v j , v k ) / ∈ E (where i < j < k < l ), then v j is called a jumping vertex for v i , v k , and v l . Forexample, the vertex v is a jumping vertex in the graph shown in Fig. 2(c). Clearly, anon-jumping labeling does not contain any jumping vertex. -Graphs and Monotone L -Graphs 5 Fig. 2. (a) A non-jumping labeling of a graph G , (b) an ordering γ = { v , v , v , v , v } of G ,(c) a jumping vertex v , (d) a jumping graph G (cid:48) , and (e) an L -embedding of G (cid:48) . One can easily verify that paths, cycles, and complete graphs are non-jumping graphs.In this section, we describe several other types of graphs can be classified as non-jumping graphs. We begin with outerplanar graphs.
Theorem 1.
Let G be an outerplanar graph. Then G is a non-jumping graph, and anon-jumping labeling of G can be found in linear time.Proof. Every outerplanar graph admits a one page book embedding [4] which can befound in linear time.In a one page book embedding of a graph, we place each ver-tex of the graph on the spine of the book and each edge can be drawn on one pagewithout edge crossing. If we consider the sequence of vertices as a labeling of a onepage book embedding, there is no jumping vertex because there is no pair of edges ( v i , v k ) , ( v j , v l ) ∈ E where i < j < k < l . (cid:117)(cid:116) We next show that (bull, dart, gem)-free chordal graphs are non-jumping graphs.The bull, dart and gem are shown in the Fig. 3(a). Before the proof, we define a fewterms as follows: Let T be a tree, and v be a vertex of T . We denote the subtree of T rooted at v by T v . We denote the parent of v by v (cid:48) , and parent of v (cid:48) by v (cid:48)(cid:48) . A vertex u issaid to be an uncle of v if u (cid:48) = v (cid:48)(cid:48) . Theorem 2.
Every (bull, dart, gem)-free chordal graph is a non-jumping graph.Proof.
Let G = ( V, E ) be a (bull, dart, gem)-free chordal graph of n vertices. Thenthere is a tree T whose leaves correspond to the vertices of G such that two vertices areadjacent in G precisely when their distance in T is at most three [26]; see Fig. 3(b-c).Hence G is a 3-leaf power graph of T . We use the notation v to indicate that the leaf v of T corresponds to the vertex v in G . Let u and v be vertices of G .Since G is a 3-leaf power graph of T , ( u, v ) ∈ E if and only if u and v are siblings,or u is an uncle of v , or v is an uncle of u . We find an ordering of vertices of G using T as follows: We first root T at a non-leaf vertex x of T . We then sort each subtree of T rooted at each vertex in counterclockwise order, according to depth in ascending order.Let γ (cid:48) = { v , v , v , . . . , v n } be the ordering of the leaves taken from the coun-terclockwise DFS traversal on T starting from x . We now prove that γ = { v , v , v , . . . , v n } is a non-jumping labeling of G by supposing that ( v i , v k ) , ( v j , v l ) ∈ E with i < j < k < l , and showing that ( v j , v k ) ∈ E . Ahmed et al.
Fig. 3. (a) A bull, a dart, a gem. (b) A 3-leaf power graph G , and (c) the 3-leaf power tree of G . It is easy to see that ( v i , v k ) ∈ E if and only if v i and v k are siblings, or v i is anuncle of v k . Since we sorted the vertices by their depth before taking the ordering, andsince i < k , v k can not be an uncle of v i . Thus, we have two cases to consider: Case 1: v i and v k are siblings. Since the order was taken from DFS traversal, v i , v i +1 , . . . , v k are siblings. So, v j and v k are siblings, and we have ( v j , v k ) ∈ E . Case 2: v i is an uncle of v k . If v i and v j are siblings, then ( v j , v k ) ∈ E , because thedistance between v i and v k and the distance between v j and v k are the same. Otherwise,we can prove that v j and v k are siblings. Suppose that v j and v k are not siblings. Then v j ∈ T o , where o is a non-leaf child of v (cid:48) i that was encountered before v (cid:48) k in the traversal.Thus, the path between v j and v l contains v (cid:48) i due to the ordering of the vertices and thepositions of i, j, k, and l ; see Fig. 11. Now, the distance between v (cid:48) i and v j is at least 2,and the distance between v (cid:48) i and v l is at least 2. This means that ( v j , v l ) / ∈ E , which isnot true. The contradiction shows that v j and v k must be siblings and ( v j , v k ) ∈ E . (cid:117)(cid:116) We now show that every interval graph has a non-jumping labeling.
Theorem 3.
Every interval graph is a non-jumping graph.
Fig. 4. (a) A graph G , (b) an interval representation of G , and (c) an L -embedding of G . Proof.
Let G be an interval graph of n vertices. Let S = {| a , b | , | a , b | , . . . , | a n , b n |} be an interval representation of G . We denote the endpoints a i and b i of the interval cor-responding to v i by v i ( a ) and v i ( b ) , respectively. Note that v i ( a ) < v i ( b ) ; see Fig. 4.Let γ = { v , v , . . . , v n } be an ordering of vertices of G in non-decreasing order of v i ( a ) (1 ≤ i ≤ n ) . If i < j then v i ( a ) < v j ( a ) . We now prove that γ is a non-jumpinglabeling of G . By way of contradiction, assume that γ is a jumping labeling. Then γ con-tains a jumping vertex v j . By definition, there exists edges ( v i , v k ) , ( v j , v l ) ∈ E with -Graphs and Monotone L -Graphs 7 i < j < k < l such that ( v j , v k ) / ∈ E This means v i ( b ) > v k ( a ) and v j ( b ) > v l ( a ) .By construction, v j ( b ) > v k ( a ) , since v k ( a ) < v l ( a ) and v l ( a ) < v j ( b ) . Since v j ( b ) > v k ( a ) and v j ( a ) < v k ( a ) , we have ( v j , v k ) ∈ E , a contradiction. (cid:117)(cid:116) Theorem 4.
Every convex bipartite graph is a non-jumping graph.Proof.
Let G = ( R ∪ B, E ) be a convex bipartite graph with V ( G ) = R ∪ B , where R ∩ B = ∅ . Without loss of generality, suppose that G is convex over B . Then thereexists a bijective mapping f : B → { , , . . . , | B |} such that for all v ∈ R and any twovertices x, y ∈ N ( v ) , there is no vertex z ∈ B \ N ( v ) such that f ( x ) < f ( z ) < f ( y ) . Fig. 5. (a) A convex bipartite graph G and (b) a vertex or-dering of G . We define s : R → B so that for any vertex v ∈ R , s ( v ) = min b ∈ N ( v ) f ( b ) . Sup-pose we sort the vertices r ∈ R in non-increasing order of s ( r ) .Let R sort be the new ordering,and let B f be the vertices b ∈ B sorted in increasing order of f ( b ) . For example, in Fig. 5(b), R sort = { e, b, c, a, d } and B f = { , , , , } . We nowprove that γ = { R sort , B f } is anon-jumping labeling of V ( G ) .For γ to be a non-jumping labeling, it must be true that for all positions i < j Theorem 5. Not all graphs are non-jumping graphs. Recall that a monotone L -embedding is an L -embedding such that the corners ofeach L -segment are on a straight line. We can completely characterize monotone L -graphs in terms of non-jumping graphs. Ahmed et al. Theorem 6. A graph G admits a monotone L -embedding if and only if G is a non-jumping graph. We prove Theorem 6 by first showing that any non-jumping graph G admits a mono-tone L -embedding in Lemma 1. The converse is proven in Lemma 2.Before we begin, we note that if a graph has a monotone L -embedding with thecorners of the L ’s on a line that is drawn vertically or horizontally, then for any pairof vertices ( v i , v j ) , there can only be an edge ( v i , v j ) if i + 1 = j , i.e., the graph isa subgraph of a path. Thus, the graph is trivially non-jumping, as there cannot be anyindices i < j < k < l in a labeling γ such that ( v i , v k ) ∈ E or ( v j , v l ) ∈ E . Agraph with no edges is also trivially non-jumping, and admits a “degenerate” monotone L -embedding in which no L would intersect another even if their arms were extendedindefinitely. v v v v y = x monotone line Fig. 6. Monotone positioning of the L -segments corresponding to v , . . . , v on the line y = x . For convenience, we define a coordinatesystem over the quarter-plane R beginningwith (0 , in the top-left corner, and x - and y -coordinates increasing to the right and down-ward respectively. This choice of coordinate sys-tem will allow us to construct a monotone L -embedding so that the corner of every L lies on theline y = x . Moreover, any non-trivial monotone L -embedding can be expanded (see Lemma 3),translated, and rescaled to create an equivalentembedding with the corners of each L on this line.Note that once we have a drawing with the cornersof each L line arranged on y = x , we can performarbitrary affine transformations on the coordinatesystem without rotating any of the L ’s themselves.For the rest of the paper, unless otherwise indi-cated, every monotone L -embedding will have its corners aligned on the line y = x inthis way. Lemma 1. Let G be a non-jumping graph of n vertices and m edges. Then G admitsmonotone L -embedding on a grid of size O ( n ) × O ( n ) , and this embedding can becomputed in O ( n + m ) time.Proof. Let G be a non-jumping graph of n vertices and γ = { v , v , . . . , v n } be a non-jumping labeling of G . If there are edges ( v i , v k ) , ( v j , v l ) ∈ E such that i < j < k < l ,then ( v j , v k ) ∈ E We now construct an L -monotone drawing for G using the coordinate system givenabove. Let L ( v ) be the L -drawing of vertex v . Then ( v.x, v.y ) is the corner of L ( v ) andthe horizontal and vertical arms of L ( v ) have lengths v.w and v.h respectively.For each v j ∈ V , let v j .x = v j .y = 2 j ; this places all corners on the line y = x .Also, for each v j , if there exists an index i < j such that some ( v i , v j ) ∈ E , then for By non-trivial, we mean a monotone L -embedding with at least one intersection, and with L ’sthat are not aligned horizontally or vertically. -Graphs and Monotone L -Graphs 9 a = min { i | i < j and ( v i , v j ) ∈ E } , define v j .h = 2 | j − a | +1 . If there is no such index i , then let v j .h = 1 . Similarly, if there is some index k > j such that ( v j , v k ) ∈ E , thenlet b = max { k | k > j and ( v j , v k ) ∈ E } and define v j .w = 2 | j − b | + 1 ; otherwise, let v j .w = 1 ; see Fig. 6.To see that this is a valid L -monotone drawing, first recall that the corners of all the L ’s are on the diagonal line y = x . Also note that for each index a < b , v a .x < v b .x and v a .y < v b .y . We must show that for indices a and b with ( v a , v b ) ∈ E , L ( v a )and L ( v b ) intersect. Without loss of generality, suppose a < b . Then, we must showthat: | v a .x − v b .x | < v a .w and | v a .y − v b .y | < v b .h ; Finally, we must show that for ( v j , v k ) / ∈ E with j < k , either | v j .x − v k .x | > v j .w or | v j .y − v k .y | > v k .h .It is clear from the computation of the width and height of each L that whenever ( v a , v b ) ∈ E , L ( v a ) and L ( v b ) intersect as described above.Now let v j and v k be vertices in G such that ( v j , v k ) / ∈ E , with j < k . Supposefor a contradiction that L ( v j ) and L ( v k ) intersect; that is, | v j .x − v k .x | < v j .w and | v j .y − v k .y | < v k .h . Since v j and v k are not adjacent, by the construction of v j .w ,there must be some vertex v l such that l > k and v j .w = | v j .x − v l .x | + 1 > | v j .x − v k .x | Similarly, by the construction of v l .h , there must be some vertex v i such that i < j and v k .h = | v k .y − v i .y | + 1 > | v j .y − v k .y | We now have i < j < k < l and the two edges ( v i , v k ) , ( v j , v l ) ∈ E . Since the indices i, j, k, l are taken from a non-jumping labeling of G , we must have ( v j , v k ) ∈ E , acontradiction.The entire drawing is contained in a rectangle of dimensions n × n . To see this,note that no corner of any L -segment will be placed to the left of the line x = 2 , norbelow the line y = 2 n . Also, no horizontal arm of an L will extend to the right beyondthe line x = 2 n + 1 , as this is one unit to the right of L ( v n ), nor will any vertical armextend above the line y = 1 .We can construct this drawing in O ( | V | + | E | ) time. First, for each v ∈ V , we plotthe corner of L ( v ) at ( v.x, v.y ) , and draw its two arms with unit length. Then, for eachedge ( v i , v j ) ∈ E with i < j , we extend the horizontal arm of L ( v i ) to have length atleast | i − j | + 1 , and extend the vertical arm of L ( v j ) to have length at least | i − j | + 1 . (cid:117)(cid:116) Lemma 2. Let L be a monotone L -embedding of a graph G . Then G is a non-jumpinggraph.Proof. Since L is monotone, the corners of each L ( v ) lie on a straight line. Let γ ( L ) = { L ( v ) , L ( v ) , . . . , L ( v n ) } be an ordering of the L -segments according to their cornerpositions from left to right. If the line on which the corners of the L ’s lie is horizontalor vertical, then as described above, the graph is a subgraph of a path, and is triviallynon-jumping. Similarly, if the corners lie on a line with negative slope, then there areno edges, so the graph is trivially non-jumping.The remaining possibility is that the corners of the L -segments lie on a line withpositive slope. In this case, for each pair of indices a < b , we have v a .x < v b .x and v a .y < v a .y . Now, γ ( L ) gives us an ordering γ = { v , v , . . . , v n } of the correspondingvertices of G . We want to prove that γ is a non-jumping labeling of G . For any fourvertices v i , v j , v k , v l with i < j < k < l , we must show that if L ( v i ) intersects L ( v k )and L ( v j ) intersects L ( v l ), then L ( v j ) and L ( v k ) also intersect.To begin, note that if L ( v i ) and L ( v k ) intersect, then | v i .x − v k .x | < v i .w and | v i .y − v k .y | < v k .h . Similarly, if L ( v j ) and L ( v l ) intersect, we have | v j .x − v l .x | < v j .w and | v j .y − v l .y | < v l .h. By the ordering of γ ( L ) , we have v i .x < v j .x < v k .x and v j .y < v k .y < v l .y . Thus, | v j .x − v k .x | < | v j .x − v l .x | < v j .w and | v j .y − v k .y | < | v i .y − v k .y | < v k .h . So L ( v j ) and L ( v k ) intersect, and ( v j , v k ) ∈ E . (cid:117)(cid:116) In the proof of Theorem 5, we show that the graph in Fig. 2(d) is a jumping graph.However, it is easy to verify that the L -embedding in Fig. 2(e) is the L -embedding ofthe jumping graph. In Theorem 6, we showed that a graph G admits a monotone L -embedding if and only if G is a non-jumping graph. This proves the following theorem: Theorem 7. Not all L -graphs are monotone L -graphs. While it is difficult to determine whether a particular graph is non-jumping, the follow-ing theorem shows that we can easily verify whether a given labeling for a graph is anon-jumping labeling. Theorem 8. Given a graph G = ( V, E ) with vertex labeling γ = { v , v , . . . , v n } , itcan be determined in O ( | V | log | V | + | E | ) time whether γ is a non-jumping labelingfor G .Proof. Using the procedure described in Lemma 1, we can construct an L -monotoneembedding of a graph G = ( V, E ) in O ( | V | + | E | ) time given a non-jumping labeling γ . Let us call the procedure P .From Lemma 2, we know that given any L -monotone embedding, we can constructa non-jumping labeling γ by sorting the vertices v i ∈ G ( V ) in increasing order of thiscorner coordinate v i .x . Let us call this order V sort . The γ thus constructed from V sort is a non-jumping labeling. P produces a valid L -monotone embedding if and only if the input labeling γ isnon-jumping. To prove this, let us suppose we get a valid L -monotone embedding from P using a jumping labeling γ jump . Let us arrange the vertices v i in increasing order ofcorner coordinates v i .x to obtain V sort . V sort must give a non-jumping labeling. Thus,our assumption that γ jump is a jumping labeling is invalid and we get a contradiction.We can use this to test if any γ is non-jumping or not. Let the drawing produced by P using γ be P ( γ ) . If P ( γ ) is a valid L -monotone embedding, γ must be non-jumping.A valid L -embedding has | V | line segments (one vertical and one horizontal for each L -shape). Similarly, there are | E | + | V | line segment intersections (one intersection forevery ( v i , v j ) ∈ E , and one intersection at the corner of each L -shape).Using an orthogonal line segment intersection search (e.g., a sweep line algorithmas described in [13]), all intersections in P ( γ ) can be listed in O ( N log N + k ) time, -Graphs and Monotone L -Graphs 11 where N = 2 | V | is the number of total line segments, and k = O ( N ) is the number ofpossible intersections. It suffices to check the first | E | + | V | intersections to determineif P ( γ ) is a valid L -monotone embedding: if there is an unwanted intersection in thefirst | E | + | V | intersections, then the embedding is invalid, and γ is jumping. On theother hand, if there are more than | E | + | V | intersections, these additional intersectionsmust be invalid and γ is jumping. Otherwise, γ is non-jumping.Since only the first k = | E | + | V | intersections need to be examined, we only need O ( | V | log | V | + | E | ) time to determine if a labeling is non-jumping or not. (cid:117)(cid:116) L -graphs In this section we prove that distance-hereditary graphs and k -leaf power graphs (for k ≤ ) admit L -embeddings. We begin with a lemma about transformations of L -embeddings; this result will allow us to derive new L -graphs from old ones. Lemma 3. Let L be an L -embedding of a graph G , and L ( v ) be the L corresponding tovertex v . Then, a valid L -embedding L (cid:48) of G can be constructed from L by expandingan infinitesimal slice of L (cid:48) that is parallel to an arm of L ( v ).Proof. There are four ways in which L can be expanded:a) Expand L rightward with respect to L ( v ) as follows: For every L ( w ) in L with acorner to the right of L ( v ), move L ( w ) to the right by one unit; also, for every vertex L ( u ) with a corner to the left of or vertically aligned with L ( u ) that intersects suchan L ( w ), extend the horizontal arm of L ( u ) to the right by one unit.b) Expand L leftward with respect to L ( v ) as follows: For every L ( u ) in L with a cornerto the left of L ( v ), move L ( u ) to the left by one unit; also, if such an L ( u ) intersectsa L ( w ) that has its corner vertically aligned with or to the right of L ( v ), then extendthe horizontal arm of L ( u ) by one unit.c) Expand L upward with respect to L ( v ), by replacing the words ‘right’ and ‘left’ inthe description of expanding rightward above with ‘up’ and ‘down’ respectively,and exchanging the words ‘horizontal’ and ‘vertical’.d) Expand L downward by similarly modifying the description of a leftwards expan-sion.To show that any of these operations produces another valid embedding L (cid:48) of G , wecan simply observe that in each transformation, all intersections of L ’s are preserved,and no new intersections are introduced. (cid:117)(cid:116) Using this result, we find that certain modifications of an L -graph result in another L -graph. Theorem 9. Let G be a graph that admits an L -embedding, and G (cid:48) be a graph con-structed from G by adding a pendant vertex, a true twin, or a false twin in G . Then G (cid:48) admits an L -embedding. Proof. Let L be an L -embedding of G , and suppose we derive G (cid:48) from G by adding apendant vertex v to G with neighbor u . Let L ( u ) represent u in L . To create L (cid:48) , we mustplace L ( v ) so that it intersects with L ( u ) and no other L . To be sure there is room to doso, we first expand L rightward two units, and both upward and downward one unit,with respect to L ( u ). We then place L ( v ) with its corner one unit to the right and oneunit below the corner of L ( u ), giving it horizontal arm length 1 and vertical arm length2. Suppose instead that we derive G (cid:48) from G by replacing a vertex u with true twinvertices v and w so that v and w are adjacent to all the neighbors of u , and are alsoadjacent to one another. We construct L (cid:48) representing G (cid:48) as follows. Replace L ( u ) with L ( v ), so that L ( v ) retains all the intersections of L ( u ). Now expand the drawing bothrightward and downward one unit with respect to L ( v ) to create room for L ( w ). We give L ( w ) a vertical arm length that is one greater than that of L ( v ), and a horizontal armlength one less than that of L ( v ) after the rightward expansion, and place its corner oneunit down and to the right of the corner of L ( v ). Thus, L ( v ) and L ( w ) each intersectevery L -segment that L ( u ) intersected, and also intersect each other.If we construct G (cid:48) from G by replacing a vertex u with false twin vertices v and w , we can proceed similarly. We first expand L leftward and downward one unit withrespect to L ( v ). Next, we place L ( w ) one unit down and to the left of L ( v ), and give L ( w ) vertical and horizontal arm lengths one unit greater than those of L ( v ). Now L ( w )intersects the same L -segments that L ( v ) intersects, but does not intersect L ( v ) itself. (cid:117)(cid:116) If G is a distance-hereditary graph, then G can be built up from a single vertex bya sequence of the following three operations: a) add a pendant vertex, b) replace anyvertex with a pair of false twins, and c) replace any vertex with a pair of true twins. [3]Thus, Theorem 9 immediately yields the following corollary. Corollary 1. Let G be a distance-hereditary graph. Then G is an L -graph. It is easy to see that -leaf power graphs and -leaf power graphs are L -graphs. Wecan use Theorem 2 and Lemma 1 to also show that -leaf power graphs are L -graphs.The proof of the following theorem on -leaf power graphs is given in the Appendix. Theorem 10. Every -leaf power graph admits an L -embedding. We have shown that several classes of graphs, such as distance-hereditary graphs and k -leaf power graphs for low values of k are L -graphs. We have also provided a completecharacterization of the more restricted variant of monotone L -graphs by correspondencewith the class of non-jumping graphs. This type of graph has a combinatorial descrip-tion, expressed as the existence of a specific type of linear order of its vertices.The results of our paper suggest several open problems: What is the complexityof determining whether a given graph G is a non-jumping? Are all planar graphs L -graphs? Are k -leaf power graphs L -graphs for k > ? Our future work will investigatethese questions. -Graphs and Monotone L -Graphs 13 References 1. Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.:String graphs of k-bend paths on a grid. Electronic Notes in Discrete Mathematics 37,141 – 146 (2011), 2. 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In this mathematical proof we use patterns that can occurin the labeling process to show that however a labeling is chosen, it is jumping. Fig. 7. Notation for the vertices of thegraph used in the proof of Theorem 5. Before beginning the formal proof, we de-fine the notation depicted in Fig. 7, to make eas-ier the identification of jumping patterns. Specif-ically, we provide each vertex with a name X zy where X ∈ { I, II, III } depending on its con-nection with other vertices: each II is connectedwith two I ’s, one II , and one III ; each III isconnected with one I and two II ’s; and the I ’s arethe remaining vertices. Next, we have y ∈ { l, r } if X ∈ { II, III } and y ∈ { in, out } if X = { I } ,such that III l ( III r ) is connected with two II l ’s( II r ’s). We also have I in connected with each II and each III , while I out is connected with each II , and I in and I out are not connected. Finally,we have z ∈ { t, b } indicating whether two II ’sare connected —i.e., two II ’s are adjacent iff theyhave the same value of z .To simplify our proof, in the following we use h, k ∈ { l, r } where h (cid:54) = k if notspecified, and i, j ∈ { t, b } where i (cid:54) = j if not specified. Hence, for two vertices II ih and II jk we have: – h = k ⇔ the vertices are adjacent to the same III . – i = j ⇔ the vertices are adjacentTo prove that every possible labeling is jumping, we first temporary remove the twovertices of type I . Their removal produces a cycle composed of six vertices.Observe that a cycle has a non-jumping representation if we fix the label of one vertexand the other vertices are labeled in sequence, or if the labels of two adjacent verticesare swapped, i.e., the labels of any pair of adjacent vertices are at distance of at mosttwo. Thus, for our cycle the feasible (i.e., non-jumping) sequences are: – III h II ih II ik III k II jk II jh , – II jh III h II ih II ik III k II jk , – II jk II jh III h II ih II ik III k ,as well as all sequences obtained by permuting exactly two consecutive vertices. (Forthe sake of brevity, in the following we consider only the sequences without such a permutation. However, this proof can easily be extended to accommodate the permutedsequences.)Before adding the two I ’s to the sequences given above, we consider the followinginfeasible, i.e. jumping, configuration. Note that we use the notation ( . . . ) to signify toany vertex or sequence of vertices: ( . . . ) II ( . . . ) I ( . . . ) I ( . . . ) II ( . . . ) (1)In general, the two I ’s cannot be placed between any two pair of vertices of type II ,since the I ’s are not connected by an edge, but must be connected to every II .From the infeasible configuration 1, it follows that at least one I should be placed tothe left (or right) of all the II ’s. Without loss of generality, we consider only placing I tothe left of the II ’s. After placing one I in this way, we have the following possibilities: – I III h II ih II ik III k II jk II jh – III h I II ih II ik III k II jk II jh – I II jh III h II ih II ik III k II jk – I II jk II jh III h II ih II ik III k We now observe that two nonadjacent II ’s cannot be placed between the two I ’s,that is, ( . . . ) I ( . . . ) II h ( . . . ) II k ( . . . ) I ( . . . ) (2)is unfeasible if h (cid:54) = k is infeasible. This is because there is no an edge between II h and II k , while there must be an edge between each I and each II . From this infeasibleconfiguration, it follows that between the two I ’s there can be only zero, one, or two II ’s.Because III k and II h do not share an edge, the sequence ( . . . ) I ( . . . ) III k ( . . . ) II h ( . . . ) II k ( . . . ) (3)is infeasible, and because II ik and II jk do not share an edge, we also have ( . . . ) I ( . . . ) II k ( . . . ) II k ( . . . ) III k ( . . . ) (4)is infeasible. As such, we can eliminate the first and the last sequences in the previouslist.The following list reports all the possible sequences with both I ’s. Here, the avail-able positions for the second I that remain after considering the previous infeasibleconfigurations are shown inside parentheses: – III h I ( I ) II ih ( I ) II ik ( I ) III k ( I ) II jk II jh – I ( I ) II jh ( I ) III h ( I ) II ih II ik III k II jk Now, if we identify the two I ’s as I in and I out , we find that the following areinfeasible configurations: ( . . . ) I in ( . . . ) I out ( . . . ) III ( . . . ) II ( . . . ) (5) -Graphs and Monotone L -Graphs 17 because I out and III do not share and edge, and ( . . . ) II ( . . . ) III h ( . . . ) I out ( . . . ) II h ( . . . ) (6)because no III and I out share an edge. Thus, II cannot be followed by a III , a I out and the adjacent II of the previous III .We observe that the previous configurations can occur in any of the two sequences.So the leftmost I can only be I out : – III h I out ( I in ) II ih ( I in ) II ik ( I in ) III k ( I in ) II jk II jh – I out ( I in ) II jh ( I in ) III h ( I in ) II ih ( I in ) II ik III k II jk We finally conclude that all the remaining sequences are jumping since they eachcontain at least one of the infeasible configurations: ( . . . ) III ( . . . ) I out ( . . . ) I in ( . . . ) II ( . . . ) (7)because I out and I in do not share an edge, and ( . . . ) I ( . . . ) II ik ( . . . ) II ih ( . . . ) III k ( . . . ) (8)because II ik and II ih do not share an edgeIt follows that the graph does not have a non-jumping labeling. As mentioned, thisproof can easily be extended to any exchange of an adjacent pair of vertices of the cycle. Proof of Theorem 10:Theorem. Every -leaf power graph admits an L -embedding. Fig. 8. (a) A tree T , which is modified by (b) removing multiple siblings, and (c) adding dummyleaves are to the internal vertices. (d) A rooted simplified leaf-tree of T . Before we begin our proof, we first define some notation and terminology. Let G =( V, E ) be a 4-leaf power graph. Then there is a tree T (cid:48) whose leaves correspond to thevertices of G in such a way that two vertices are adjacent in G precisely when theirdistance in T (cid:48) is at most 4. Let v be a vertex of G . We denote the graph obtained byremoving v and all edges incident to v by G − v . To make T (cid:48) as a simpler and more uniform tree, we remove and add some leaveson T (cid:48) as follows. We first remove siblings leaves from T (cid:48) , i.e., leaves that have thesame parent, and add new leaves to each internal node without any child-leaf to createa rooted simplified leaf-tree T of G from T (cid:48) , as follows. Let u and v be two siblingsleaves in T (cid:48) . Then N ( u ) = N ( v ) in G and ( u, v ) ∈ E . Hence, u is a true twin of v in G . According to Theorem 9, if we have L ( G − v ) then we have L ( G ). For each groupof sibling leaves, we keep exactly one leaf, removing the others. For example, Figure8(b) shows the transformed tree after removing multiple siblings from the tree shownin Fig. 8(a). We now add a dummy leaf to the internal vertices of T that do not havea child-leaf. In Figure 8(c), vertices y and z are dummy leaves. The dummy verticeswill be removed from L ( G ) after the construction of L -embedding of G . We make T arooted tree by selecting an arbitrary internal vertex as its root (see Fig. 8(d)). Observethat every internal vertex has exactly one leaf in the rooted simplified leaf-tree. Fig. 9. A fully connected L -embedding Let v be a vertex of T . We denote thesubtree rooted at v by T v . We denote theparent of v by v (cid:48) , and the parent of v (cid:48) by v (cid:48)(cid:48) . Vertices u and v are said to be sib-lings if u (cid:48) = v (cid:48) . A vertex u is said tobe an uncle of v if u (cid:48) = v (cid:48)(cid:48) . A vertex u is said to be a p-uncle of v if u (cid:48) is par-ent of v (cid:48)(cid:48) . Vertices u and v are said to be cousins if u (cid:48)(cid:48) = v (cid:48)(cid:48) . A vertex u is said tobe a nephew of v if u (cid:48)(cid:48) = v (cid:48) . The lengthof the largest path from the root of T toany leaf is called depth of T .Since G is a 4-leaf power graph, then ( u, v ) ∈ E if and only if one of the fol-lowing is true:1. u and v are siblings,2. u and v are cousins,3. u is an uncle of v or v is an uncle of u , or4. u is a p-uncle of v or v is a p-uncle of u .By definition, a leaf u has exactly one uncle and exactly one p-uncle. Also, cousins havea common uncle and a common p-uncle.We now draw each group of cousins as a fully connected L -embedding in which allof the L -segments intersect one another (see Fig. 9).Let r be the root of T . Since T is a rooted simplified leaf-tree, r has exactly onechild-leaf, which we call r c . We denote a nephew of r c by r cc , and the subtree inducedby the vertices r , r c , and r cc by T conf .We maintain two configurations, a " Rectangle-configuration " and an " L-configuration ,"for the drawing of leaves of T conf of T . The properties of a Rectangle-configuration are: RconPro1 The horizontal arms of cousins intersect the vertical arm of their uncle,maintaining a subdivided L-shaped free region for each cousin -Graphs and Monotone L -Graphs 19 RconPro2 The vertical part of each subdivided L-shaped free region is visible fromthe horizontal arm of corresponding cousin RconPro3 The horizontal part of every subdivided L-shaped free region is visible fromthe vertical arm of the uncleThe properties of an L-configuration are the following: LconPro1 The vertical segments of cousins intersect the horizontal segment of theiruncle, maintaining a subdivided rectangle shape-free region for each cousin LconPro2 The right part of each subdivided rectangle shape-free region is visible fromthe vertical arm of its corresponding cousin LconPro3 The top part of every subdivided rectangle shape-free region is visible fromthe horizontal arm of the uncleFigure 10(f) depicts a Rectangle-configuration for the tree drawn in Fig. 10(d), andFig. 10(e) depicts an L-configuration for the tree drawn in Fig. 10(d).We need the following lemma for the proof of Theorem 10 Lemma 4. Let G be a 4-leaf power graph and T be a corresponding rooted simplifiedleaf-tree of G . Then T conf admits a Rectangle-configuration and an L-configuration.Proof. Let r be the root of T . Assume that T conf of T contains k + 1 leaves. Theseare r c and k nephews of r c (for some k ≥ ); the k nephews are cousins of eachother. We find a Rectangle-configuration of T conf as follows. We take a fully connected L -embedding of the k cousins and add L ( r c ) such that the horizontal arms of the k cousins intersect the vertical arm of L ( r c ). It is easy to verify that the properties of theRectangle-configuration hold (see Fig. 10(b)). Similarly, we find L-configuration of T as follows. We take a fully connected L -embedding of k cousins and add L ( r c ) suchthat the vertical arms of the k cousins intersect the horizontal segment of L ( r c ). Thisdrawing maintains the properties of L-configuration of T (see Fig. 10(c)). Proof.