Finding the skeletons of the cubic 9-cages with EG3 (extended abstract)
Dan Ashlock David L. Schweizery May 1, 1998 Abstract
A complete list of the cubic nine-cages was presented in [7]; the list was generated by computer and no drawings were provided. In this paper, we describe methods for drawing many of those graphs so as to maximize the displayed symmetries. These heuristics and the software used to implement them (the graph theory package EG3) may be extended to other graph drawing problems, particularly including the dicult problem of drawing graphs of high girth (such as the cubic 11-cage, which is one of the problems in this year's graph drawing competition).
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Introduction and Motivation
The problem of the (3,9)-cages [16, 6, 11] was in many repects solved in [7], where a complete list of all eighteen cubic graphs of girth nine on 58 vertices was presented. The list was obtained by a pruned exhaustive search carried out by computer, however, with the result that the presentations were limited to adjacency lists. Even with the additional information presented (such as the size and generators of the automorphism group), the graphs remain visually inaccessible. All eighteen graphs are Hamiltonian, so [7] gave their adjacencies as the chords to be added to a cycle through 58 vertices. This suggest drawing the graphs as 58 points on a circle and 29 chords. Doing so with the most symmetrical of the graphs, for example, which has an automorphism group of order 24, yields the picture in Figure 1 At around the same time, the present authors were working on a structural approach to the cubic cages [2] and, at the twenty-third Southeastern International Conference on Combinatorics, Graph Theory, and Computing, presented the drawing of a cubic graph of girth nine on 58 vertices shown in Figure 2 (in which solid and open vertices with corresponding labels are identi ed). Department of Mathematics, Iowa State University, Ames, IA, 50010, email:
[email protected] y Department of Mathematics, College of the Holy Cross, Worcester, MA, 01610, email:
[email protected] (Principal contact for inquiries about this paper.)
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Figure 1: Graph G3 from [7]
a5
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Figure 2: A 58 vertex cubic graph of girth 9 2
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Figure 3: Cubic trees
Figure 4: The skeleton portion of Figure 2, redrawn These two graphs are isomorphic; the central questions addressed in this paper are whether and how a drawing like the second, which displays many of the symmetries of the graph, may be created from the rst. Insight into the structural and symmetry properties of the graph in Figure 2 is primarily available from the part on the left. Therefore, throughout this paper, we work from the heuristic assumption that it is useful to analyze each of these graphs as a combination of a cubic forest and another, smaller graph. We call this smaller graph the skeleton of the cage; the cubic forest will be omitted from the drawings but will be described as a combination of cubic trees as notated in Figure 3. The graph in Figure 2 was created by hand, which resulted in a more symmetrical (and hence more informative) drawing of the skeleton being overlooked initially. That skeleton can also be presented as a truncated cube, as in Figure 4. The methods of this paper, applied to this graph as presented in Figure 1 (i.e., using only the information available from [7] and the assumption that we are removing a cubic forest), yield either of the skeletons in Figure 5. The rst of those is obviously the truncated cube; the second is the skeleton found by removing four copies of t3 : 3
Figure 5: Two skeletons of the graph in Figure 1
Figure 6: Disjoint t3s: centers at distance at least ve
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Skeletons
2.1 Heuristic Methods
In order to decompose a cubic nine-cage into a skeleton and a cubic forest, it is necessary to nd several vertex-disjoint cubic trees in the cage. This task is simpli ed by observing that it suces to nd either widely separated vertices (if one is looking for copies of t3 ) or edges (if one is looking for copies of t4). Figure 6 indicates that vertices at distance ve or more lie at the center of disjoint copies of t3; gure 7 indicates that vertices of the line graph at distance six or more correspond to edges at the center of disjoint copies of t4 (the open circles indicate the vertices of the line graph). In general, this method can be more formally described as the problem of nding a code with large minimum distance in the shortest-path metric of the graph or its line graph; the vertices of edges so found are the centers of cubic trees that may be deleted to yield a pleasing skeleton. Recall that the mth power [15, ex. 6.2.17, p. 228] of a graph G; Gm ; is obtained by adding edges to G between all pairs of vertices at mutual distance m or less; de ne the closely related mth distance of a graph to be the graph on 4
Figure 7: Disjoint t4 s: center edges at distance at least six the same vertex set with edges between all pairs of vertices at distance exactly m: Vertices at distance exactly k may be found by looking for adjacent vertices in the kth distance of the cage, and vertices at distance at least k may be found by looking for adjacent vertices in the complement of the k ? 1st power. (These ideas extend to edges simply by using the line graph.) Finally, collections of mutually disjoint cubic trees may be found by looking for cliques in those graphs. EG3 provides functions for facilitating all these operations.
2.2 A Complete Example
We now turn to a step-by-step narrative description of the use of EG3 to nd the skeletons of the graph G3, starting from only the information available in Figure 1. The reader is strongly encouraged to download a copy of EG31 and follow the process \live". EG3 is command-driven via menus; menu items can be selected by cursor or by hot-key. In the description below, the capitalized letter in each menu selection is the hot-key for that item. Submenu items are written immediately following the menu choice from which they descend, separated by a slash: Edit/Draw/Springs means choose Edit from the main menu, then Draw from the edit menu, then Springs from the drawing menu. Some commands cause EG3 to prompt the user for additional input. 1. Read Enter g3, with path information if necessary to locate the le g3.gdf. 2. New/Z - clone current EG3 creates a copy of the currently selected graph, and makes that copy the selected graph. 3. Edit/Build graphs/Line graph EG3 builds the line graph. 4. [Edit/Build]/dIstance Enter 6. EG3 builds the 6th distance of the graph. Note that once the line graph has been built, EG3 leaves control at the Edit/Build graph menu, so only one keystroke is required to invoke a new graph building operation. 1 EG3 is available from http://www.math.iastate.edu/danwell/eg.html; it will run on almost any IBM-compatible PC. Data les for a number of interesting graphs, including the cubic nine-cages, are also available there.
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5. [Edit/Build]/Done Pop up one menu level, back to the Edit menu. 6. [Edit]/Invariants/K - clique nder Enter 3. EG3 seeks a 3-clique (i.e., a copy of K3) in the graph. It indicates the 3-clique it nds by coloring the vertices, in this case, the vertices 1{58, 41{53, and 47{48 (each of which is named in the obvious fashion for the edge of the original graph to which it corresponds). The clique nder is randomized, so running it again is likely to nd a di erent clique, if one exists. (This is the only 3-clique in this graph.) Make a note of the vertices in the clique. 7. [Edit/Invariants]/Done Return to the Edit menu. 8. [Edit]/Quit Return to the main menu without saving changes to the graph being edited. In this case, the graph will revert to its original form | the line graph is not saved. 9. Edit/Draw/dIstance Enter 1 58. EG3 draws the graph with vertices 1 and 58 at the top of the screen and the rest of the vertices organized into rows by their smaller distance to 1 or 58. Thus, all vertices adjacent to either 1 or 58 are drawn in the rst level down, all vertices adjacent to any of those are drawn in the next level down, and so on. 10. [Edit/Draw]/Done 11. [Edit]/Color edges/Incident Enter 1 2 9 36 57 and select red from the pop-up color bar. EG3 colors every edge incident on any of the listed edges red. This colors one of the copies of t4 : 12. [Edit/Color edges]/Done 13. [Edit]/Draw/dIstance Enter 41 53. 14. [Edit/Draw]/Done 15. [Edit]/Color edges/Incident Enter 41 40 42 52 54 and select yellow from the pop-up color bar. 16. [Edit/Color edges]/Done 17. [Edit]/Draw/dIstance Enter 47 48. 18. [Edit/Draw]/Done 19. [Edit]/Color edges/Incident Enter 47 12 19 46 49 and select green from the pop-up color bar. 20. [Edit/Color edges]/Done We now have the graph with three vertexdisjoint copies of t4 clearly marked. 21. [Edit]/Save Return to the main menu, saving the changes to the graph. 22. New/Z - clone current 6
23. Edit/toggle Edges/By color Select green from the pop-up color bar. EG3 removes all green edges from the graph. 24. [Edit/toggle Edges]/By color Select yellow from the pop-up color bar. 25. [Edit/toggle Edges]/By color Select red from the pop-up color bar. 26. [Edit/toggle Edges]/Quit 27. [Edit]/Kill vertices/by deGree Enter 0. EG3 removes all vertices of degree 0. (These are the vertices that were completely internal to the copies of t4:) 28. [Edit/Kill vertices]/Done 29. [Edit]/Save We now have the skeleton, but we still need to create a symmetrical drawing. 30. New/Z - clone current 31. Edit/Build graphs/Two vanish EG3 elides vertices of degree 2. This smoothing operation is often quite useful. 32. [Edit/Build]/Done 33. [Edit]/Draw/Springs EG3 draws the graph as if the vertices obeyed an inverse-square repulsive force and the edges were Hooke's law springs. (The physical parameters may be adjusted by the user.) Hit d when the drawing has stabilized. 34. [Edit/Draw]/Normalize EG3 resize the graph to t the screen. Make a note of the vertices around the outer 9-cycle, which will probably be 33 32 29 28 27 6 5 4 34. 35. [Edit/Draw]/Done 36. [Edit]/Save 37. (Use the left-arrow key to reselect the complete skeleton.) 38. Edit/Draw/Clean Enter the list of vertices from the 9-cycle found previously (i.e., 33 32 29 28 27 6 5 4 34). EG3 uses an algorithm due to Nate Dean, xing the speci ed vertices at equally spaced points around a circle and then using an averaging algorithm to place the rest. 39. [Edit/Draw]/Done 40. [Edit]/Save The collection of graphs may now be saved to a le if desired. The Edit/Postscript menu can be used to dump EPS format versions of the graphs for inclusion in printed materials. 7
Figure 8: The skeleton of the graph G18
2.3 Catalog of Skeletons
We now turn to some speci c examples of the type of structural information that can be obtained about the cubic 9-cages via the methods of this paper. Many of the graphs of [7] are rigid, that is, have an automorphism group of order 1, and hence have skeletons without symmetries. Several of the graphs do not admit a decomposition into a forest of four t3s or three t4s; they require heuristics discussed here only in one example (graph G15). 2.3.1
G18
We begin with the last graph listed in [7], G18, because of the striking visual similarity to our original starting point. This graph, which has an automorphism group of order 6, may be decomposed into three t4s and the skeleton shown in Figure 8. The similarity of this graph to the graph on the left side of 5 is quite unmistakable and shows a clear structural relationship between the two graphs. 2.3.2
G14
Similarly, G14, which has an automorphism group of order 12, also yields a familiar-appearing skeleton. In this case, we may decompose the graph into four t3s and the skeleton of Figure 9. This graph is similar to, but less symmetrical than, the graph on the right side of Figure 5. 2.3.3
G4 and G5
Just as the skeleton of G3 may be viewed as being derived from a truncated cube (as in Figure 4), graphs G4 and G5 are structurally related in that both have skeletons derived from a hexagonal prism with two corners truncated. In G4, each six-sided face has one truncated corner; in G5, both truncated corners 8
Figure 9: The skeleton of the graph G14
Figure 10: The reduced skeletons of the graphs G4 and G5 lie on the same face. For clarity, these prisms are shown in Figure 10 with the degree two vertices elided; the corresponding complete skeletons are shown in Figure 11. 2.3.4
G15
The graph G15 is one of the more symmetrical of the cubic 9-cages, with an automorphism group of order 8. However, it admits neither of the skeleton decompositions discussed so far. A little investigation with EG3 reveals that a single t7 may be removed, leaving the skeleton shown in Figure 12 (the reduced skeleton is shown to highlight the hexagonal structure).
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Figure 11: The skeletons of the graphs G4 and G5
Figure 12: The skeleton of the graph G15
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Figure 13: The skeleton of the graph G10 2.3.5
G10
Graph G10 has an automorphism group of order 2: the skeleton cannot show much symmetry. Figure 13 shows the skeleton, con rming both the symmetry of order 2 and the lack of other symmetries.
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EG3
EG3 is a versatile and easy-to-use tool for research and teaching in graph theory. It is able to compute a number of graph-valued functions (power, distance, line graph, and complement have been illustrated above), color vertices and edges either manually or following various graph-theoretic properties, compute useful invariants of the graph, draw the graph using several algorithms (manual, spring physics, Hamiltonize, etc.), alter the graph following graph-theoretic information, and write PostScript.
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Conclusions and Future Directions
Heuristic methods combined with graph drawing software can yield signi cant structural insight into otherwise dicult graphs. In the case of the cubic 9cages, the heuristics are suciently regular to be able to be applied to the entire 11
collection of graphs in an almost mechanical manner but are also suciently general to extend to other cubic cages. Possibilities for the future include: apply these approaches to other cages, invert these sorts of structural decompositions as a guide to automated searches for larger cubic cages, and revise and upgrade EG3.
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