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The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper…
We show that every cubic bridgeless graph G has at least 2^(|V(G)|/3656) perfect matchings. This confirms an old conjecture of Lovasz and Plummer. This version of the paper uses a different definition of a burl from the journal version of…
An st-path is a path with the end-vertices s and t. An s-path is a path with an end-vertex s. The results of this paper include necessary and sufficient conditions for a {claw, net}-free graph G with given two different vertices s, t and an…
We introduce loose graph simulations (LGS), a new notion about labelled graphs which subsumes in an intuitive and natural way subgraph isomorphism (SGI), regular language pattern matching (RLPM) and graph simulation (GS). Being a…
Operator learning has been highly successful for continuous mappings between infinite-dimensional spaces, such as PDE solution operators. However, many operators of interest-including differential operators-are discontinuous or set-valued,…
In many real-world applications, it is undesirable to drastically change the problem solution after a small perturbation in the input, as unstable outputs can lead to costly transaction fees, privacy and security concerns, reduced user…
Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,...,v_{\nu}\}$, which may have multiple edges but have no loops, and $2\leq d_G(v_i)\leq 3$ for $i=1,2,...,\nu$, where $d_G(v)$ denotes the degree of vertex $v$ of $G$. We show…
Given a graph $G=([n],E)$ and $w\in\R^E$, consider the integer program ${\max}_{x\in \{\pm 1\}^n} \sum_{ij \in E} w_{ij}x_ix_j$ and its canonical semidefinite programming relaxation ${\max} \sum_{ij \in E} w_{ij}v_i^Tv_j$, where the maximum…
In a semi-supervised learning scenario, (possibly noisy) partially observed labels are used as input to train a classifier, in order to assign labels to unclassified samples. In this paper, we study this classifier learning problem from a…
For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain $H$ as an induced subgraph. Already in the early 1980s Alekseev observed that if $H$ is connected, then the \textsc{Max Weight Independent Set} problem (MWIS)…
The stability number of a graph G, is the cardinality of a stable set of maximum size in G. If the stability number of G remains the same upon the addition of any edge, then G is called $\alpha ^{+}$-stable. G is a K\"{o}nig-Egervary graph…
Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $\Theta_2^{\text{P}}$-complete. They studied the common graph parameters…
Let $\A$ and $\B$ be operator algebras with $c_0$-isomorphic diagonals and let $\K$ denote the compact operators. We show that if $\A\otimes\K$ and $\B\otimes\K$ are isometrically isomorphic, then $\A$ and $\B$ are isometrically isomorphic.…
For a graph $G=(V,E)$, a subset $D$ of vertex set $V$, is a dominating set of $G$ if every vertex not in $D$ is adjacent to atleast one vertex of $D$. A dominating set $D$ of a graph $G$ with no isolated vertices is called a paired…
The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs, and is further and deeply developed in…
A general (convex) polytope $P\subset\mathbb R^d$ and its edge-graph $G_P$ can have very distinct symmetry properties. We construct a coloring (of the vertices and edges) of the edge-graph so that the combinatorial symmetry group of the…
Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality \[ \hom(G, H…
In this paper, we aim to design sparse D-optimal (determinantoptimal) pose-graph SLAM problems through the synthesis of sparse graphs with the maximum weighted number of spanning trees. Characterizing graphs with the maximum number of…
Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its…
A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph…