Related papers: 2-switch: transition and stability on forests and …

Given any two forests with the same degree sequence, we show in an algorithmic way that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests. We…

In the present work we prove that given any two unicycle graphs (pseudoforests) that share the same degree sequence there is a finite sequence of 2-switches transforming one into the other such that all the graphs in the sequence are also…

A 2-switch is an edge addition/deletion operation that changes adjacencies in the graph while preserving the degree of each vertex. A well known result states that graphs with the same degree sequence may be changed into each other via…

In this work, we delve into the study of the 2-switch-degree of a graph $G$, which is nothing more than the degree of $G$ as a vertex of the realization graph $\mathcal{G}(d)$ associated with the degree sequence $d$ of $G$. We explore the…

We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main…

We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},\ldots,C_{k+1}$ such that there is an $i$ such that for…

Given a graph, we can form a spanning forest by first sorting the edges in some order, and then only keep edges incident to a vertex which is not incident to any previous edge. The resulting forest is dependent on the ordering of the edges,…

We prove that for $k \in \mathbb{N}$ and $d \leq 2k+2$, if a graph has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests, where one of the pseudoforests has all connected components having…

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the…

A 2-stab unit interval graph (2SUIG) is an axes-parallel unit square intersection graph where the unit squares intersect either of the two fixed lines parallel to the $X$-axis, distance $1 + \epsilon$ ($0 < \epsilon < 1$) apart. This family…

The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge…

We develop the theory of transformation semigroups that have degree 2, that is, act by partial functions on a finite set such that the inverse image of points have at most two elements. We show that the graph of fibers of such an action…

A graph G is a 2-tree if G=K_3, or G has a vertex v of degree 2, whose neighbours are adjacent, and G\v{i}s a 2-tree. A characterization of the degree sequences of 2-trees is given. This characterization yields a linear-time algorithm for…

Let $G$ be a graph and $T_1,T_2$ be two spanning trees of $G$. We say that $T_1$ can be transformed into $T_2$ via an edge flip if there exist two edges $e \in T_1$ and $f$ in $T_2$ such that $T_2= (T_1 \setminus e) \cup f$. Since spanning…

Let $N_2DL(v)$ denote the set of degrees of vertices at distance 2 from $v$. The $2$-neighborhood degree list of a graph is a listing of $N_2DL(v)$ for every vertex $v$. A degree restricted $2$-switch on edges $v_1v_2$ and $w_1w_2$, where…

The existence of a partition of the common set of the vertices of two forests into two subsets, when difference of their capacities in the neighborhood of each vertex of each forest not greater than 2 is proved, and an example, which shows…

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups…

A graph G is a multi-interval PCG if there exist an edge weighted tree T with non-negative real values and disjoint intervals of the non-negative real half-line such that each node of G is uniquely associated to a leaf of T and there is an…

We investigate relations between the pseudo-orbit-tracing property, topological stability and openness for tree-shifts. We prove that a tree-shift is of finite type if and only if it has the pseudo-orbit-tracing property which implies that…

We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an…