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Register a new userSimple expressions for the long walk distance
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Added by
Pavel Chebotarev
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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The walk distances in graphs are defined as the result of appropriate transformations of the $sum_{k=0}^infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a connected weighted graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic, moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. In this paper, simple expressions for the long walk distance are obtained. They involve the generalized inverse, minors, and inverses of submatrices of the symmetric irreducible singular M-matrix ${cal L}=rho I-A,$ where $rho$ is the Perron root of $A.$
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By rectangle packing we mean putting a set of rectangles into an enclosing rectangle, without any overlapping. We begin with perfect rectangle packing problems, then prove two continuity properties for parallel rectangle packing problems, and discuss how they might be used to obtain negative results for perfect rectangle packing problems.
A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) to mathbb{Q}_{geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $X$ such that $G-X$ is a split graph. It is easy to show that a graph is a split graph if and only it it does not contain a $4$-cycle, $5$-cycle, or a two edge matching as an induced subgraph. Therefore, SVD admits an easy $5$-approximation algorithm. On the other hand, for every $delta >0$, SVD does not admit a $(2-delta)$-approximation algorithm, unless P=NP or the Unique Games Conjecture fails. For every $epsilon >0$, Lokshtanov, Misra, Panolan, Philip, and Saurabh recently gave a randomized $(2+epsilon)$-approximation algorithm for SVD. In this work we give an extremely simple deterministic $(2+epsilon)$-approximation algorithm for SVD.
We prove that there exists a function $f(k)=mathcal{O}(k^2 log k)$ such that for every $C_4$-free graph $G$ and every $k in mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. This answers a question of Kim and Kwon [ErdH{o}s-Posa property of chordless cycles and its applications. JCTB 2020].
$H_q(n,d)$ is defined as the graph with vertex set ${mathbb Z}_q^n$ and where two vertices are adjacent if their Hamming distance is at least $d$. The chromatic number of these graphs is presented for various sets of parameters $(q,n,d)$. For the $4$-colorings of the graphs $H_2(n,n-1)$ a notion of robustness is introduced. It is based on the tolerance of swapping colors along an edge without destroying properness of the coloring. An explicit description of the maximally robust $4$-colorings of $H_2(n,n-1)$ is presented.
A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach.
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