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.

/pdf/a-mathematical-bibliography-of-signed-and-gain-graphs-and-3a49f10hs2.pdf

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

The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

The Graph Crossing Number and its Variants: A Survey

Embeddings of graphs with no short noncontractible cycles

We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another? We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straight-line segments). We highlight the similarities and differences of the two settings, describe many extensions and generalizations, highlight algorithmic issues, outline several applications and mention open problems.

Flips in planar graphs

Uniqueness and faithfulness of embedding of toroidal graphs

Diagonal flips in triangulations of surfaces

/pdf/diagonal-transformations-of-graphs-on-closed-surfaces-236a7zn0gs.pdf

Diagonal Transformations of Graphs on Closed Surfaces

An embedding f: G —► R of a graph G into R is said to be linear if each edge f(e) (e G E(G)) is a straight line segment. It will be shown that for any knot or link type k , there is a finite number R(k) such that every linear embedding of the complete graph Kn with at least R(k) vertices (n > R(k)) in R contains a knot or link equivalent to k .

/pdf/ramsey-theorems-for-knots-links-and-spatial-graphs-2646awx0ll.pdf

Ramsey theorems for knots, links and spatial graphs

/pdf/irreducible-triangulations-of-the-klein-bottle-24iwsnwjfn.pdf

Seiya Negami

Papers

Uniqueness and faithfulness of embedding of toroidal graphs

Diagonal flips in triangulations of surfaces

Diagonal Transformations of Graphs on Closed Surfaces

Ramsey theorems for knots, links and spatial graphs

Irreducible Triangulations of the Klein Bottle