Linear Operators That Preserve the Genus of a Graph

Mathematics

If a graph can be embedded in a smooth orientable surface of genus g without edge crossings and can not be embedded on one of genus g − 1 without edge crossings, then we say that the graph has genus g. We consider a mapping on the set of graphs with m vertices into itself. The mapping is called a linear operator if it preserves a union of graphs and it also preserves the empty graph. On the set of graphs with m vertices, we consider and investigate those linear operators which map graphs of genus g to graphs of genus g and graphs of genus g + j to graphs of genus g + j for j ≤ g and m sufficiently large. We show that such linear operators are necessarily vertex permutations.

Linear and Multilinear Algebra, 2018

A graph has genus k if it can be embedded without edge crossings on a smooth orientable surface of genus k and not on one of genus k−1. A mapping of the set of graphs on n vertices to itself is called a linear operator if the image of a union of graphs is the union of their images and that maps the edgeless graph to the edgeless graph. We investigate linear operators on the set of graphs on n vertices that map graphs of genus k to graphs of genus k and graphs of genus not k to graphs of genus not k. We show that such linear operators are necessarily vertex permutations.

Discrete Mathematics, 1989

It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.

Journal of Combinatorial Theory, Series B, 2006

In 1976, Stahl and White conjectured that the nonorientable genus of Kl,m,n, where l ≥ m ≥ n, is (l−2)(m+n−2) 2 ¡ . The authors recently showed that the graphs K3,3,3 , K4,4,1, and K4,4,3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.

2008 49th Annual IEEE Symposium on Foundations of Computer Science, 2008

For every fixed surface S, orientable or non-orientable, and a given graph G, Mohar (STOC'96 and Siam J. Discrete Math. (1999)) described a linear time algorithm which yields either an embedding of G in S or a minor of G which is not embeddable in S and is minimal with this property. That algorithm, however, needs a lot of lemmas which spanned six additional papers. In this paper, we give a new linear time algorithm for the same problem. The advantages of our algorithm are the following: 1. The proof is considerably simpler: it needs only about 10 pages, and some results (with rather accessible proofs) from graph minors theory, while Mohar's original algorithm and its proof occupy more than 100 pages in total. 2. The hidden constant (depending on the genus g of the surface S) is much smaller. It is single exponential in g, while it is doubly exponential in Mohar's algorithm. As a spinoff of our main result, we give another linear time algorithm, which is of independent interest. This algorithm computes the genus and constructs minimum genus embeddings of graphs of bounded tree-width. This resolves a conjecture by Neil Robertson and solves one of the most annoying long standing open question about complexity of algorithms on graphs of bounded tree-width.

Quantum Topology, 2013

For a graph embedded into a surface, we relate many combinatorial parameters of the cycle matroid of the graph and the bond matroid of the dual graph with the topological parameters of the embedding. This will give an expression of the polynomial, defined by M. Las Vergnas in a combinatorial way using matroids as a specialization of the Krushkal polynomial, defined using the symplectic structure in the first homology group of the surface.

Transactions of the American Mathematical Society, 1970

With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph K2m,2m with itself is l + 8m2(m-1). Furthermore, let ßi" be the graph K",s and recursively define the cartesian product ßi," = ß?L x x Klfl for nä2. The genus of ß("" is shown to be 1 + 2" " 3s"(sn-4), for all n, and í even ; or for n > 1, and s = 1 or 3. The graph ßi,1' is the 1-skeleton of the «-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles. Introduction. In this paper a graph G is a finite 1-complex. The genus y(G) of G is the minimum genus among the genera of all compact orientable 2-manifolds in which G can be imbedded. All 2-manifolds in this paper are assumed to be compact and orientable. There are very few families of graphs for which the genus has been determined; these include the complete graphs (Ringel and Youngs [7]), the complete bipartite graphs, (Ringel [5]), and some subfamilies of the family of complete tripartite graphs (see [6] and [8]). One of the first genus formulae was developed by Ringel [4] in 1955 (and independently by Beineke and Harary [1] in 1965) when he found that the genus of the M-cube Qn is given by: y(Qn) = l+2"-3(n-4), for n ^ 2. The «-cube can be defined as a repeated cartesian product: let Qx = K2, the complete graph on two vertices, and recursively define Qn = Qn _ x x K2 for n ^ 2. In general, given two graphs Gx and G2, with vertex sets V(GX), V(G2) and edge sets E(GX), E(G2) respectively, the cartesian product Gx x G2 is formed by taking V(GX x G2) ={(ux, u2) : ux e V(GX), u2 e V(G2)} and E(GX x G2)={[(ux, u2), (vx, v2)]: ux = vx

Ars Combinatoria Waterloo Then Winnipeg, 1997

The star graph S n is a graph with S n the set of all permutations over f1; : : :; ng as its vertex set; two vertices 1 and 2 are connected if 1 can be obtained form 2 by swapping the rst element of 1 with one of the other n ? 1 elements. In this paper we establish the genus of the star graph. We show that the genus, g n of S n , is exactly equal to n!(n?4)=6+1 by establishing a lower bound and inductively giving a drawing on a surface of appropriate genus.

The well-known Euler characteristic is an invariant of graphs defined by means of the vertex, edge and face numbers of a graph, to determine the genus of the underlying surface of the graph. By means of it, it is possible to determine the vertex, edge and face numbers of all possible graphs which can be drawn in a given orientable/non-orientable surface. In this paper, by means of a given degree sequence, a new number denoted by () G Ω which is related to Euler characteristic and has several applications in Graph Theory is defined. This formula gives direct information compared with the Euler characteristic on the realizability, number of realizations, connectedness, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc.

Journal of Research of the National Bureau of Standards Section B Mathematics and Mathematical Physics, 1965

iFe br ua ry 3, 1964) TH EOR EM : A 1·1 co rres po nd e nce be t wee n the ed ge s of two co nn ec ted g ra ph s is a uu a lit y with re s pe c t to s u me po lyh edral s urface e mb e ddin g if and o nl y if for ea c h ve rte x v uf e ac h gra ph , th e ed ges whi ch mee t v co rres po nd in th e oth e r graph to th e ed ge s of a s ub graph G,· whi c h is co nn ec ted a nd whi c h has a n e ve n numb e r of it s e d ge·e nd s to e ac h of it s ve rti ces (w he re if a n ed ge mee ts va t bo th e nd s it s ima ge in G,. is co unt ed twi c e) . Us in g th e Eul e r furmul a, th e charac te ri s ti c of the s urfa ce is de termin ed by th e two gra ph s. Thu s, th e th eore m ge ne rali zes a var ia ti o n of th e H. Whitn ey conditi o n fur a gra ph to be pl a nar.