21,177 research outputs found
On some special classes of contact -VPG graphs
A graph is a -VPG graph if one can associate a path on a rectangular
grid with each vertex such that two vertices are adjacent if and only if the
corresponding paths intersect at at least one grid-point. A graph is a
contact -VPG graph if it is a -VPG graph admitting a representation
with no two paths crossing and no two paths sharing an edge of the grid. In
this paper, we present a minimal forbidden induced subgraph characterisation of
contact -VPG graphs within four special graph classes: chordal graphs,
tree-cographs, -tidy graphs and -free graphs. Moreover, we present a
polynomial-time algorithm for recognising chordal contact -VPG graphs.Comment: 34 pages, 15 figure
Acyclic edge coloring of graphs
An {\em acyclic edge coloring} of a graph is a proper edge coloring such
that the subgraph induced by any two color classes is a linear forest (an
acyclic graph with maximum degree at most two). The {\em acyclic chromatic
index} \chiup_{a}'(G) of a graph is the least number of colors needed in
an acyclic edge coloring of . Fiam\v{c}\'{i}k (1978) conjectured that
\chiup_{a}'(G) \leq \Delta(G) + 2, where is the maximum degree of
. This conjecture is well known as Acyclic Edge Coloring Conjecture (AECC).
A graph with maximum degree at most is {\em
-deletion-minimal} if \chiup_{a}'(G) > \kappa and \chiup_{a}'(H)
\leq \kappa for every proper subgraph of . The purpose of this paper is
to provide many structural lemmas on -deletion-minimal graphs. By using
the structural lemmas, we firstly prove that AECC is true for the graphs with
maximum average degree less than four (\autoref{NMAD4}). We secondly prove that
AECC is true for the planar graphs without triangles adjacent to cycles of
length at most four, with an additional condition that every -cycle has at
most three edges contained in triangles (\autoref{NoAdjacent}), from which we
can conclude some known results as corollaries. We thirdly prove that every
planar graph without intersecting triangles satisfies \chiup_{a}'(G) \leq
\Delta(G) + 3 (\autoref{NoIntersect}). Finally, we consider one extreme case
and prove it: if is a graph with and all the
-vertices are independent, then \chiup_{a}'(G) = \Delta(G). We hope
the structural lemmas will shed some light on the acyclic edge coloring
problems.Comment: 19 page
Deploying robots with two sensors in -free graphs
Let be a graph of minimum degree at least two with no induced subgraph
isomorphic to . We prove that if is not isomorphic to one of eight
exceptional graphs, then it is possible to assign two-element subsets of
to the vertices of in such a way that for every
and every vertex the label is assigned to
or one of its neighbors. It follows that has fractional domatic number
at least . This is motivated by a problem in robotics and generalizes a
result of Fujita, Yamashita and Kameda who proved that the same conclusion
holds for all -regular graphs
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
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