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2-Switch Transition on Unicyclic Graphs and Pseudoforest

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2-Switch Transition on Unicyclic Graphs and Pseudoforest 2-switch transition on unicyclic graphs and pseudoforest Daniel A. Jaumea,∗, Adr´ıan Pastinea, Victor N. Schv¨ollnera a Departamento de Matem´aticas, Facultad de Ciencias F´ısico-Matem´aticas y Naturales, Universidad Nacional de San Luis, Instituto de Matem´atica Aplicada San Luis (UNSL-CONICET), Universidad Nacional de San Luis, San Luis, Rep´ublica Argentina. Abstract In the present work we prove that given any two unicycle graphs (pseud- oforests) 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 unicyclic graphs (pseudoforests). Keywords: 2-switch, degree sequence, unicyclic graphs, pseudoforests, stability, Berge family. 2000 MSC: 05A15, 05A19. 1. Introduction Every graph G = (V, E) in the present article is finite, simple, undirected and labeled. We use |G| and kGk to denote the order of G and the size of G respectively. In this work the set of vertices of G is always a subset of [n] := {1,...,n}, for some n. When there may be ambiguity we use V (G) and E(G) to denote the vertex set and the edge set of G, otherwise we just use V and E. Vertex adjacency is denoted by x ∼ y, and we denote the edge {x,y} by xy (i.e., we say that xy ∈ E). The number of connected components of a graph G is denoted by κ(G) and its set of components by K(G). The subgraph of G obtained by deleting vertex v is denoted by G−v. Similarly, G − e is the subgraph of G obtained by deleting edge e, G + e or arXiv:2103.00618v1 [math.CO] 28 Feb 2021 G+ab is the graph obtained by adding an edge to G. If W is a set of vertices (edges) of a graph G, G − W denotes the subgraph obtained by deleting the vertices (edges) in W . The degree sequence of a graph G with vertex set V (G) = [n] is the sequence s(G) = (d1,...,dn), where di is the degree of vertex i. We assume ∗ Corresponding author: Daniel A. Jaume Email addresses: [email protected] (Daniel A. Jaume), [email protected] (Adr´ıan Pastine), [email protected] (Victor N. Schv¨ollner) Preprint submitted to Elsevier Received: date / Accepted: date that d1 ≥ d2 ≥···≥ dn. A sequence s = (d1,...,dn) is graphical if there is a graph such that s is its degree sequence. Let s = (d1,...,dn) be a graphical sequence. By U(s) we denote the set of all the unicyclic graphs (connected graphs with just one cycle) with degree sequence s, and by P(s) we denote the set of all the pseudoforests (graphs whose connected components are trees or unicyclic graphs) with degree sequence s. The name pseudoforest is justified by analogy to the more commonly studied trees and forests. In [4] the authors attribute the study of pseud- oforests to Dantzig’s 1963 book on linear programming, see [3], in which pseudoforests arise in the solution of certain network flow problems. Pseud- oforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs (their number of edges is linearly bounded in terms of their number of vertices, in fact, they have at most as many edges as they have vertices) and their matroid struc- ture allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests. The name “pseudoforest” was first used in [12]. One of the most studied problems in the literature is: given a graph parameter (clique number, domination number, matching number, etc.), finding the minimum and maximum values for the parameter in a family of graphs, see [2, 6, 7, 11]. Another interesting problem is deciding which values between the minimum and the maximum can be realized by a graph in the family, see [9, 10, 8]. Let G be a graph containing four distinct vertices a, b, c, d such that ab, cd ∈ E and ac, bd∈ / E. The process of deleting the edges ab and cd from G and adding ac and bd to G is referred to as a 2-switch in G, this is a classical operation, see [1]. Even though the order of the vertices matters, it is usual to talk about a 2-switch between the edges {a, b} and {c, d}. If G′ is the graph obtained from G by a 2-switch, it is straightforward to check that G and G′ have the same degree sequence. In other words, this operation preserves the degree sequence. In [8] the 2-switch was introduced as a function. Definition 1.1. ([8]) Let a, b, c, d ∈ [n] and let G be a graph. The ma- a b trix c d is said to be interchangeable in G, if it satisfies the following conditions: 1. ab, cd ∈ E(G); 2. {a, b} ∩ {c, d} = ∅; 3. ac, bd 6∈ E(G). a b Otherwise, c d is said to be trivial for G. 2 Notice in particular that if at least one of a, b, c, d is not a vertex of G, a b then c d is trivial for G. a b Definition 1.2. ([8]) Let n be a integer and a, b, c, d ∈ [n], A = c d and G a graph. A 2-switch is a function τA : G → G, where G is the set of all graphs, defined as follows: G − ab − cd + ac + bd, if A is interchangeable in G, τA(G)= (1) G, if A is trivial for G. If τA(G)= G, we say that τA is trivial for G. The matrix A is said to be an action matrix of τA. An important fact about degree sequences is that, given two graphs with the same degree sequence, one can be obtained from the other by applying successive 2-switches. Theorem 1.3. If G, H ∈ G(s), there exists a 2-switch sequence transforming G into H. Theorem 1.3 appears throughout the literature, although its earliest ref- erence appears most likely in [1]. In [2] the authors study the matching number of trees with a given degree sequence. The authors find minimum and maximum values for the matching number in this family, and then show that every value between the minimum and the maximum is realized by a graph in the family. When this happens for a parameter, it is said to have interval property with respect to the family of graphs being studied. A family of graphs is said to be a Berge family, if for all pair of graphs of the family, there exist a finite sequence of 2-switch that transform one into the other, such that every intermediate graph is a member of the family. In [8] it was proved that trees and forests with the same degree sequence are Berge families, they also introduce two special types of 2-switch. A nontrivial 2-switch τ over a tree T is said to be a t-switch if τ(T ) is a tree. A nontrivial 2-switch τ over a forest F is said to be an f-switch if τ(F ) is a forest. They also proved the following theorem that shows that trees and forests with the same degree sequence are Berge families. Theorem 1.4 (Forest Transition Theorem, [8]). Let F and F ′ be two forests (trees) with the same degree sequence. Then F can be transformed into F ′ by a finite sequence of f-switches (t-switches). A graph parameter ξ is said to be stable under 2-switch, see definition 5.1 in [8], if given G a graph and τ a 2-switch, then |ξ (τ(G)) − ξ(G)|≤ 1. 3 Theorem 1.5 ([8]). The following parameters are stable under 2-switch: 1. matching number, 2. independence number, 3. domination number, 4. path-covering number, 5. edge-covering number, 6. vertex-covering number, 7. chromatic number, 8. clique number, 9. number of connected components. Note that any stable discrete graph parameter has the interval property with respect to every Berge family. 2. u-switch In this section we introduce and characterize a particular type of 2- switch, the u-switch, which is closed over unicyclic graphs. We start with some technical results. A graph G is said to be unicyclic if it is connected and contains exactly one cycle, i.e. kGk = |G|. If s is a graphical sequence, we denote by U(s) the set of all unicyclic graphs with degree sequence s. We assume that s is the degree sequence of at least one unicyclic graph. A 2-switch τ is said to be a breaker over a graph G whenever κ(τ(G)) > κ(G). If C is a cycle graph, and τ is a breaker over C, then τ(C) is the union of two disjoint cycles. If τ is not a breaker over C, then τ(C) is a cycle isomorphic to C. a b The following observation will be used many times. Let τ = c d be a 2-switch and e an edge of G, e∈ / {ab, cd}. If e∈ / τ(G−e), then τ(G−e)+ e = τ(G). In particular, if τ(G − e) is a tree, then τ(G) is unicyclic. We define two edge-disjoint subgraphs of a graph G: Cycles(G) and Forest(G). By Cycles(G) we denote the subgraph induced by all the ver- tices of G such that they are in a cycle of G.
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