Edmonds, Lovász, and Pulleyblank showed that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut. Carvalho et al. gave a stronger conjecture: if a matching covered graph has a nontrivial tight cut $C$, then it also has a nontrivial ELP-cut that does not cross $C$. Chen, et al gave a proof of the conjecture. This note is inspired by the paper of Carvalho et al. We give a simplified proof of the conjecture, and prove the following result which is slightly stronger than the conjecture: if a nontrivial tight cut $C$ of a matching covered graph $G$ is not an ELP-cut, then there is a sequence $G_1=G, G_2,\ldots,G_r, r\geq2$ of matching covered graphs, such that for $i=1, 2,\ldots, r-1$, $G_i$ has an ELP-cut $C_i$, and $G_{i+1}$ is a $C_i$-contraction of $G_i$, and $C$ is a $2$-separation cut of $G_r$.
An edge cut $C$ of a graph $G$ is {\it tight} if $|C \cap M|=1$ for every perfect matching $M$ of $G$.~Barrier cuts and 2-separation cuts are called {\it ELP-cuts}, which are two important types of tight cuts in matching covered graphs.~Edmonds, Lovász and Pulleyblank proved that if a matching covered graph has a nontrivial tight cut, then it also has a nontrivial ELP-cut.~Carvalho, Lucchesi, and Murty made a stronger conjecture: given any nontrivial tight cut $C$ in a matching covered graph $G$, there exists a nontrivial ELP-cut $D$ in $G$ which does not cross $C$.~We confirm the conjecture i…
The perfect matching polytope, i.e. the convex hull of (incidence vectors of) perfect matchings of a graph is used in many combinatorial algorithms. Kotzig, Lovász and Plummer developed a decomposition theory for graphs with perfect matchings and their corresponding polytopes known as the tight cut decomposition which breaks down every graph into a number of indecomposable graphs, so called bricks. For many properties that are of interest on graphs with perfect matchings, including the description of the perfect matching polytope, it suffices to consider these bricks. A key result by Lovász on…
A {\em brick} is a non-bipartite matching covered graph without non-trivial tight cuts. Bricks are building blocks of matching covered graphs. We say that an edge $e$ in a brick $G$ is {\em $b$-invariant} if $G-e$ is matching covered and a tight cut decomposition of $G-e$ contains exactly one brick. A 2-edge-connected cubic graph is {\em essentially 4-edge-connected} if it does not contain nontrivial 3-cuts. A brick $G$ is {\em near-bipartite} if it has a pair of edges $\{e_1, e_2\}$ such that $G-\{e_1,e_2\}$ is bipartite and matching covered. Kothari, de Carvalho, Lucchesi and Little proved t…
A connected graph $G$, of order two or more, is matching covered if each edge lies in some \pema. The tight cut decomposition of a matching covered graph $G$ yields a list of bricks and braces; as per a theorem of Lov{á}sz~\cite{lova87}, this list is unique (up to multiple edges); $b(G)$ denotes the number of bricks, and $c_4(G)$ denotes the number of braces that are isomorphic to the cycle $C_4$ (up to multiple edges). Two edges $e$ and $f$ are mutually dependent if, for each perfect matching $M$, $e \in M$ if and only if $f \in M$; Carvalho, Lucchesi and Murty investigated this notion in the…
A graph $G$ is called fractional $[a,b]$-covered if for every edge $e$ of $G$ there is a fractional $[a,b]$-factor with the indicator function $h$ such that $h(e)=1$. In this paper, we provide tight spectral radius conditions for graphs being fractional $[a,b]$-covered.
This paper is the first from a series of papers that provide a characterization of maximum packings of $T$-cuts in bipartite graphs. Given a connected graph, a set $T$ of an even number of vertices, and a minimum $T$-join, an edge weighting can be defined, from which distances between vertices can be defined. Furthermore, given a specified vertex called root, vertices can be classified according to their distances from the root, and this classification of vertices can be used to define a family of subgraphs called {\em distance components}. Sebö provided a theorem that revealed a relationship …
Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Turán number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$. Keevash and Sudakov proved that $f(n,H)=ex(n,H)$ if $H$ is an edge-critical graph or $C_4$ and asked if this equality holds for any graph $H$. All known exact values of this question require $H$ to contain at least one cycle. In this paper we focus on acyclic graphs and have the following results: (1) We prove $f(n,H)=ex(n,H)$ when $H$ is a spider or a double b…
A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum degree and large order to be $k$-path-coverable.
Given a set of paths $P$ we define the \emph{Path Covering with Forest Number} of $P$} (PCFN($P$)) as the minimum size of a set $F$ of forests satisfying that every path in $P$ is contained in at least one forest in $F$. We show that PCFN($P$) is treatable when $P$ is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if $P$ is a set of non-crossing shortest paths of a planar graph $G$ whose extremal vertices lie on the same face of $G$, then PCFN($P$)\leq 4$, and this bound is tight.
A polynomial bound on the pathwidth of graphs edge-coverable by $k$ shortest paths
University of Lille, CNRS, Centrale Lille, UMR 9189 CRIStAL（里尔大学、法国国家科学研究中心、中央理工-里尔学院、CRIStAL实验室）；Université Claude Bernard Lyon 1, CNRS, INSA Lyon, LIRIS, UMR 5205（里昂第一大学、法国国家科学研究中心、里昂国立应用科学学院、LIRIS实验室）；Faculty of Mathematics and Computer Science, Jagiellonian University（雅盖隆大学数学与计算机科学学院）；IRIF, Université Paris Cité（巴黎西岱大学IRIF研究所）；LaBRI, Université de Bordeaux（波尔多大学LaBRI实验室）
We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5ω(G)}{4}\rceil$ and $\lceil\frac{Δ(G) + ω(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or …
A graph $G(V,E)$ is a threshold graph if there exist non-negative reals $w_v, v \in V$ and $t$ such that for every $U \subseteq V$, $\sum_{v \in U} w_v\leq t$ if and only if $U$ is a stable set. The {\it threshold dimension} of a graph $G(V,E)$, denoted as $t(G)$, is the smallest integer $k$ such that $E$ can be covered by $k$ threshold spanning subgraphs of $G$. A permutation graph is a graph that can be represented as the intersection graph of a family of line segments that connect two parallel lines in the Euclidean plane. In this paper we will show that if $G$ is a permutation graph then $…
We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-…
