A signed graph is a graph whose edges are labeled by signs. This is a bibliography of signed graphs and related mathematics. Several kinds of labelled graph have been called "signed" yet are mathematically very different. I distinguish four types: Group-signed graphs: the edge labels are elements of a 2-element group and are multiplied around a polygon (or along any walk). Among the natural generalizations are larger groups and vertex signs. Sign-colored graphs, in which the edges are labelled from a two-element set that is acted upon by the sign group: - interchanges labels, + leaves them unchanged. This is the kind of "signed graph" found in knot theory. The natural generalization is to more colors and more general groups —  or no group. Weighted graphs, in which the edge labels are the elements +1 and -1 of the integers or another additive domain. Weights behave like numbers, not signs; thus I regard work on weighted graphs as outside the scope of the bibliography — ex cept (to some extent) when the author calls the weights "signs". Labelled graphs where the labels have no structure or properties but are called "signs" for any or no reason.

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A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas

In many applications of graph algorithms, including communication networks, graphics, assembly
planning, and VLSI design, graphs are subject to discrete changes, such as additions or deletions of
edges or vertices. In the last decades there has been a growing interest in such dynamically changing
graphs, and a whole body of algorithms and data structures for dynamic graphs has been discovered.
This chapter is intended as an overview of this ﬁeld.
In a typical dynamicgraphproblemonewould like to answerqueries ongraphs that areundergoinga sequence of updates, for instance, insertions and deletions of edges and vertices. The goal of a
dynamic graph algorithm is to update eﬃciently the solution of a problem after dynamic changes,
rather than having to recompute it from scratch each time. Given their powerful versatility, it is not
surprising that dynamic algorithms and dynamic data structures are often more diﬃcult to design
and analyze than their static counterparts.
We can classify dynamic graph problems according to the types of updates allowed. A problemis said to be fully dynamic if the update operations include unrestricted insertions and deletions of
edges. A problem is called partially dynamic if only one type of update, either insertions or deletions,
is allowed. If only insertions are allowed, the problem is called incremental; if only deletions are
allowed it is called decremental.
In this chapter, we describe fully dynamic algorithms for graph problems. We present dynamicalgorithms for undirected graphs in Section 9.2. Dynamic algorithms for directed graphs are next
described in Section 9.3. Finally, in Section 9.4 we describe some open problems.

/pdf/dynamic-graph-algorithms-2e8qry0avf.pdf

Dynamic graph algorithms

A $(k, g)$-cage is a $k$-regular graph of girth $g$ of minimum order. In this survey, we present the results of over 50 years of searches for cages. We present the important theorems, list all the known cages, compile tables of current record holders, and describe in some detail most of the relevant constructions.

http://emis.matem.unam.mx/journals/EJC/Surveys/ds16.pdf

Dynamic Cage Survey

The problem of lifting graph automorphisms along covering projections and the analysis of lifted groups is considered in a purely combinatorial setting. The main tools employed are: (1) a systematic use of the fundamental groupoid; (2) unification of ordinary, relative and permutation voltage constructions into the concept of a voltage space; (3) various kinds of invariance of voltage spaces relative to automorphism groups; and (4) investigation of geometry of the lifted actions by means of transversals over a localization set. Some applications of these results to regular maps on surfaces are given. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful re-examination of the whole subject and at the same time leads to simplification and generalization of several known results.

Lifting Graph Automorphisms by Voltage Assignments

An introduction to the theory of groups

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex v is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incident with v . Such a regular embedding can be realised on an orientable surface if and only if the group G has a subgroup H of index 2 such that H v is the cyclic subgroup of index 2 in G v . An analogous result is proved for orientably-regular embeddings.

Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces

A subgroupG of automorphisms of a graphX is said to be 12- transitive if it is vertex- and edge- but not arc-transitive. The graphX is said to be 12-transitive if AutX is 12-transitive. The correspondence between regular maps and 12-transitive group actions on graphs of valency 4 is studied via the well known concept of medial graphs. Among others it is proved that under certain general conditions imposed on a map, its medial graph must be a 12-transitive graph of valency 4 and, vice versa, under certain conditions imposed on the vertex stabilizer, a 12-transitive graph of valency 4 gives rise to\break an irreflexible regular map. This way infinite families of 12-transitive graphs are constructed from known examples of regular maps. Conversely, known constructions of 12-transitive graphs of valency 4 give rise to new examples of irreflexible regular maps. In the end, the concept of a symmetric genus of a 12-transitive graph of valency 4 is introduced. In particular, 12-transitive graphs of valency 4 and small symmetric genuses are discussed.

Maps and Half-transitive Graphs of Valency 4

According to M Gardner [“Mathematical Games: Snarks, Boojums, and Other Conjectures Related to the Four-Color-Map Theorem,” Scientific American, vol 234 (1976), pp 126–130], a snark is a nontrivial cubic graph whose edges cannot be properly colored by three colors The problem of what “nontrivial” means is implicitly or explicitly present in most papers on snarks, and is the main motivation of the present paper Our approach to the discussion is based on the following observation If G is a snark with a k-edge-cut producing components G1 and G2, then either one of G1 and G2 is not 3-edge-colorable, or by adding a “small” number of vertices to either component one can obtain snarks G1 and G2 whose order does not exceed that of G The two situations lead to a definition of a k-reduction and k-decomposition of G Snarks that for m < k do not admit m-reductions, m-decompositions, or both are k-irreducible, k-indecomposable, and k-simple, respectively The irreducibility, indecomposability, and simplicity provide natural measures of nontriviality of snarks closely related to cyclic connectivity The present paper is devoted to a detailed investigation of these invariants The results give a complete characterization of irreducible snarks and characterizations of k-simple snarks for k ≤ 6 A number of problems that have arisen in this research conclude the paper © 1996 John Wiley & Sons, Inc

Roman Nedela

Papers

Lifting Graph Automorphisms by Voltage Assignments

Characterisation of Graphs which Underlie Regular Maps on Closed Surfaces

Maps and Half-transitive Graphs of Valency 4

Decompositions and reductions of snarks

Regular Embeddings of Canonical Double Coverings of Graphs